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Номер 1, страница 4 - гдз по алгебре 10 класс учебник Абылкасымова, Жумагулова
Авторы: Абылкасымова А. Е., Жумагулова З. А.
Тип: Учебник
Издательство: Мектеп
Год издания: 2019 - 2026
ISBN: 978-601-07-1142-6
Утверждено Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 10 классе
Упражнения для повторения курса алгебры 7—9 классов - номер 1, страница 4.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113591 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "4" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-1" "field_display_title" => "1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ 0 => App\Models\Branch {#1045 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113551 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Упражнения для повторения курса алгебры 7—9 классов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "4" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1077 #items: array:1 [ 0 => App\Models\Book {#1048 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4293 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1050 #items: array:1 [ 0 => App\Models\Term {#1049 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1051 #items: array:1 [ 0 => App\Models\Term {#1052 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1053 #items: array:1 [ 0 => App\Models\Term {#1054 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #original: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1055 #items: array:2 [ 0 => App\Models\Term {#1056 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 6997 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абылкасымова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алма" "field_patronymic" => "Есимбековна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6997 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абылкасымова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алма" "field_patronymic" => "Есимбековна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1057 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 6999 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумагулова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Зауре" "field_patronymic" => "Абдыкеновна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6999 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумагулова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Зауре" "field_patronymic" => "Абдыкеновна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1058 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1059 #items: array:1 [ 0 => App\Models\Term {#1060 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1061 #items: array:1 [ 0 => App\Models\Term {#1062 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1063 #items: array:1 [ 0 => App\Models\Term {#1064 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1065 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1066 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1067 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1069 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1070 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 10 класса общественно-гуманитарного направления общеобразовательных школ" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "514" "field_priority" => "2" "field_default_folder" => "/algebra_10/Abylkasymova-u/" "field_isbn" => "978-601-07-1142-6" "field_cover" => array:1 [ 0 => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1072 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1073 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/abylkasymova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1074 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #original: array:50 [ "id" => 4293 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1050} "field_class" => Illuminate\Database\Eloquent\Collection {#1051} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1053} "field_author" => Illuminate\Database\Eloquent\Collection {#1055} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1058} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1059} "field_country" => Illuminate\Database\Eloquent\Collection {#1061} "field_city" => Illuminate\Database\Eloquent\Collection {#1063} "field_series" => Illuminate\Database\Eloquent\Collection {#1065} "field_umk" => Illuminate\Database\Eloquent\Collection {#1066} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1067} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1068} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1069} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1070} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 10 класса общественно-гуманитарного направления общеобразовательных школ" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "514" "field_priority" => "2" "field_default_folder" => "/algebra_10/Abylkasymova-u/" "field_isbn" => "978-601-07-1142-6" "field_cover" => array:1 [ 0 => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1071} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1072} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1073} "field_url" => "/10-klass/algebra/abylkasymova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1074} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1075 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1113551 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Упражнения для повторения курса алгебры 7—9 классов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047} "field_page_start" => "4" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1077} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1075} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1038 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1042 #items: array:3 [ 0 => App\Models\Element {#1082 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1290651 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1091 #items: array:1 [ 0 => App\Models\Edition {#1083 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4958 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1085 #items: array:1 [ 0 => App\Models\Term {#1084 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ …6] "field_translit" => "mektep" ] #original: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ …6] "field_translit" => "mektep" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1086 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1087 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1088 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4958 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1085} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1086} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1087} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1088} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1113591" "type" => "task" ] "text" => "<p><strong>1.</strong> Упростите выражение:</p><p><strong>а)</strong> $\frac{3-2b}{a+b} : \frac{9b+9a}{4b^2-9} - \frac{6b}{2b+3}$;</p><p><strong>б)</strong> $\frac{16a^2-25c^2}{ac+c^2} : \frac{5c-4a}{3c} + \frac{13a+14c}{a+c}$;</p><p><strong>В)</strong> $(\frac{b+2}{b^2-3b+9} - \frac{6}{b^3+27}) : \frac{3b+15}{2b^2-6b+18}$;</p><p><strong>Г)</strong> $(\frac{a-1}{a^2+2a+4} - \frac{2}{a^3-8}) : \frac{3-a}{a^2+2a+4}$;</p><p><strong>Д)</strong> $\frac{2a+1}{a-7} + (\frac{a}{a+8} + \frac{a}{7-a}) : \frac{a}{8+a}$;</p><p><strong>е)</strong> $(\frac{b}{2b-3} - \frac{b}{3+2b}) : \frac{b}{9+12b+4b^2} + \frac{45-6b}{3-2b}$;</p><p><strong>ж)</strong> $25\sqrt{b} - 0.5\sqrt{4b} + 100\sqrt{0.16b}$;</p><p><strong>3)</strong> $\sqrt{98a} + 3\sqrt{242a} - 17\sqrt{512a}$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "1-1.jpg" "alt" => null "width" => "1740" "height" => 1625 "path" => "/media/algebra_10/Abylkasymova-u/0-00/1-1.webp?ts=1753262978" ] ] ] #original: array:7 [ "id" => 1290651 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1091} "task" => array:2 [ "refs" => "1113591" "type" => "task" ] "text" => "<p><strong>1.</strong> Упростите выражение:</p><p><strong>а)</strong> $\frac{3-2b}{a+b} : \frac{9b+9a}{4b^2-9} - \frac{6b}{2b+3}$;</p><p><strong>б)</strong> $\frac{16a^2-25c^2}{ac+c^2} : \frac{5c-4a}{3c} + \frac{13a+14c}{a+c}$;</p><p><strong>В)</strong> $(\frac{b+2}{b^2-3b+9} - \frac{6}{b^3+27}) : \frac{3b+15}{2b^2-6b+18}$;</p><p><strong>Г)</strong> $(\frac{a-1}{a^2+2a+4} - \frac{2}{a^3-8}) : \frac{3-a}{a^2+2a+4}$;</p><p><strong>Д)</strong> $\frac{2a+1}{a-7} + (\frac{a}{a+8} + \frac{a}{7-a}) : \frac{a}{8+a}$;</p><p><strong>е)</strong> $(\frac{b}{2b-3} - \frac{b}{3+2b}) : \frac{b}{9+12b+4b^2} + \frac{45-6b}{3-2b}$;</p><p><strong>ж)</strong> $25\sqrt{b} - 0.5\sqrt{4b} + 100\sqrt{0.16b}$;</p><p><strong>3)</strong> $\sqrt{98a} + 3\sqrt{242a} - 17\sqrt{512a}$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "1-1.jpg" "alt" => null "width" => "1740" "height" => 1625 "path" => "/media/algebra_10/Abylkasymova-u/0-00/1-1.webp?ts=1753262978" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1089 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1291165 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1099 #items: array:1 [ 0 => App\Models\Edition {#1090 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4959 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1093 #items: array:1 [ 0 => App\Models\Term {#1092 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6714 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Gdz" "field_cases" => array:6 [ …6] "field_translit" => "gdz" ] #original: array:6 [ "id" => 6714 "created_at" => "2026-04-10 13:58:26" …4 ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1094 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1095 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1096 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4959 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1093} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1094} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1095} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1096} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1113591" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "1-1.jpg" "alt" => null "width" => "1975" "height" => 3274 "path" => "/media/algebra_10/Abylkasymova-u/1-00/1-1.webp?ts=1753263138" ] 1 => array:5 [ "name" => "1-2.jpg" "alt" => null "width" => "1955" "height" => 3513 "path" => "/media/algebra_10/Abylkasymova-u/1-00/1-2.webp?ts=1753263138" ] 2 => array:5 [ "name" => "1-3.jpg" "alt" => null "width" => "1924" "height" => 1191 "path" => "/media/algebra_10/Abylkasymova-u/1-00/1-3.webp?ts=1753263138" ] ] ] #original: array:6 [ "id" => 1291165 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1099} "task" => array:2 [ "refs" => "1113591" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "1-1.jpg" "alt" => null "width" => "1975" "height" => 3274 "path" => "/media/algebra_10/Abylkasymova-u/1-00/1-1.webp?ts=1753263138" ] 1 => array:5 [ "name" => "1-2.jpg" "alt" => null "width" => "1955" "height" => 3513 "path" => "/media/algebra_10/Abylkasymova-u/1-00/1-2.webp?ts=1753263138" ] 2 => array:5 [ "name" => "1-3.jpg" "alt" => null "width" => "1924" "height" => 1191 "path" => "/media/algebra_10/Abylkasymova-u/1-00/1-3.webp?ts=1753263138" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1097 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1549294 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107 #items: array:1 [ 0 => App\Models\Edition {#1098 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5655 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1101 #items: array:1 [ 0 => App\Models\Term {#1100 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1102 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1103 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1104 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5655 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1101} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1102} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1103} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1104} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1113591" "type" => "task" ] "text" => "<p><strong>а)</strong> $\frac{3-2b}{a+b} \cdot \frac{9b+9a}{4b^2-9} - \frac{6b}{2b+3}$</p><p>Сначала упростим первое слагаемое. Разложим числитель второй дроби и знаменатель на множители: $9b+9a=9(a+b)$ и $4b^2-9=(2b-3)(2b+3)$. Также заметим, что $3-2b=-(2b-3)$.</p><p>$\frac{-(2b-3)}{a+b} \cdot \frac{9(a+b)}{(2b-3)(2b+3)} - \frac{6b}{2b+3}$</p><p>Сократим общие множители $(a+b)$ и $(2b-3)$:</p><p>$\frac{-1}{1} \cdot \frac{9}{2b+3} - \frac{6b}{2b+3} = -\frac{9}{2b+3} - \frac{6b}{2b+3}$</p><p>Теперь вычтем дроби с одинаковым знаменателем:</p><p>$\frac{-9-6b}{2b+3} = \frac{-3(3+2b)}{2b+3} = -3$</p><p><strong>Ответ:</strong> $-3$</p><p><strong>б)</strong> $\frac{16a^2 - 25c^2}{ac+c^2} : \frac{5c-4a}{3c} + \frac{13a+14c}{a+c}$</p><p>Выполним сначала деление. Для этого разложим на множители числитель и знаменатель первой дроби: $16a^2-25c^2=(4a-5c)(4a+5c)$ и $ac+c^2=c(a+c)$. Заметим, что $5c-4a=-(4a-5c)$. Деление заменяем на умножение на обратную дробь:</p><p>$\frac{(4a-5c)(4a+5c)}{c(a+c)} \cdot \frac{3c}{-(4a-5c)} + \frac{13a+14c}{a+c}$</p><p>Сократим общие множители $c$ и $(4a-5c)$:</p><p>$\frac{4a+5c}{a+c} \cdot \frac{3}{-1} + \frac{13a+14c}{a+c} = \frac{-3(4a+5c)}{a+c} + \frac{13a+14c}{a+c} = \frac{-12a-15c}{a+c} + \frac{13a+14c}{a+c}$</p><p>Сложим дроби с одинаковым знаменателем:</p><p>$\frac{-12a-15c+13a+14c}{a+c} = \frac{a-c}{a+c}$</p><p><strong>Ответ:</strong> $\frac{a-c}{a+c}$</p><p><strong>в)</strong> $(\frac{b+2}{b^2-3b+9} - \frac{6}{b^3+27}) : \frac{3b+15}{2b^2-6b+18}$</p><p>Упростим выражение в скобках. Используем формулу суммы кубов: $b^3+27=b^3+3^3=(b+3)(b^2-3b+9)$. Приведем дроби к общему знаменателю:</p><p>$\frac{(b+2)(b+3)}{(b+3)(b^2-3b+9)} - \frac{6}{(b+3)(b^2-3b+9)} = \frac{b^2+3b+2b+6-6}{(b+3)(b^2-3b+9)} = \frac{b^2+5b}{b^3+27} = \frac{b(b+5)}{b^3+27}$</p><p>Теперь выполним деление. Разложим на множители делитель: $3b+15=3(b+5)$ и $2b^2-6b+18=2(b^2-3b+9)$.</p><p>$\frac{b(b+5)}{b^3+27} : \frac{3(b+5)}{2(b^2-3b+9)} = \frac{b(b+5)}{(b+3)(b^2-3b+9)} \cdot \frac{2(b^2-3b+9)}{3(b+5)}$</p><p>Сократим общие множители $(b+5)$ и $(b^2-3b+9)$:</p><p>$\frac{b}{b+3} \cdot \frac{2}{3} = \frac{2b}{3(b+3)}$</p><p><strong>Ответ:</strong> $\frac{2b}{3(b+3)}$</p><p><strong>г)</strong> $(\frac{a-1}{a^2+2a+4} - \frac{2}{a^3-8}) : \frac{3-a}{a^2+2a+4}$</p><p>Упростим выражение в скобках, используя формулу разности кубов: $a^3-8=a^3-2^3=(a-2)(a^2+2a+4)$.</p><p>$\frac{(a-1)(a-2)}{(a-2)(a^2+2a+4)} - \frac{2}{(a-2)(a^2+2a+4)} = \frac{a^2-2a-a+2-2}{(a-2)(a^2+2a+4)} = \frac{a^2-3a}{a^3-8} = \frac{a(a-3)}{a^3-8}$</p><p>Теперь выполним деление. Заметим, что $3-a=-(a-3)$.</p><p>$\frac{a(a-3)}{a^3-8} : \frac{-(a-3)}{a^2+2a+4} = \frac{a(a-3)}{(a-2)(a^2+2a+4)} \cdot \frac{a^2+2a+4}{-(a-3)}$</p><p>Сократим общие множители $(a-3)$ и $(a^2+2a+4)$:</p><p>$\frac{a}{a-2} \cdot \frac{1}{-1} = -\frac{a}{a-2}$</p><p><strong>Ответ:</strong> $-\frac{a}{a-2}$</p><p><strong>д)</strong> $\frac{2a+1}{a-7} + (\frac{a}{a+8} + \frac{a}{7-a}) : \frac{a}{8+a}$</p><p>Сначала выполним действия в скобках: $7-a=-(a-7)$.</p><p>$\frac{a}{a+8} + \frac{a}{-(a-7)} = \frac{a}{a+8} - \frac{a}{a-7} = \frac{a(a-7)-a(a+8)}{(a+8)(a-7)} = \frac{a^2-7a-a^2-8a}{(a+8)(a-7)} = \frac{-15a}{(a+8)(a-7)}$</p><p>Теперь выполним деление:</p><p>$\frac{-15a}{(a+8)(a-7)} : \frac{a}{a+8} = \frac{-15a}{(a+8)(a-7)} \cdot \frac{a+8}{a}$</p><p>Сократим общие множители $a$ и $(a+8)$: $\frac{-15}{a-7}$.</p><p>Теперь выполним сложение:</p><p>$\frac{2a+1}{a-7} + \frac{-15}{a-7} = \frac{2a+1-15}{a-7} = \frac{2a-14}{a-7} = \frac{2(a-7)}{a-7} = 2$</p><p><strong>Ответ:</strong> $2$</p><p><strong>е)</strong> $(\frac{b}{2b-3} - \frac{b}{3+2b}) : \frac{b}{9+12b+4b^2} + \frac{45-6b}{3-2b}$</p><p>Упростим выражение в скобках:</p><p>$\frac{b}{2b-3} - \frac{b}{2b+3} = \frac{b(2b+3)-b(2b-3)}{(2b-3)(2b+3)} = \frac{2b^2+3b-2b^2+3b}{4b^2-9} = \frac{6b}{4b^2-9}$</p><p>Теперь деление. Знаменатель делителя $9+12b+4b^2 = (3+2b)^2$.</p><p>$\frac{6b}{4b^2-9} : \frac{b}{(2b+3)^2} = \frac{6b}{(2b-3)(2b+3)} \cdot \frac{(2b+3)^2}{b}$</p><p>Сокращаем $b$ и $(2b+3)$: $\frac{6(2b+3)}{2b-3} = \frac{12b+18}{2b-3}$.</p><p>Теперь сложение. Заметим, что $3-2b=-(2b-3)$.</p><p>$\frac{12b+18}{2b-3} + \frac{45-6b}{-(2b-3)} = \frac{12b+18}{2b-3} - \frac{45-6b}{2b-3} = \frac{12b+18-(45-6b)}{2b-3} = \frac{12b+18-45+6b}{2b-3} = \frac{18b-27}{2b-3} = \frac{9(2b-3)}{2b-3} = 9$</p><p><strong>Ответ:</strong> $9$</p><p><strong>ж)</strong> $25\sqrt{b} - 0,5\sqrt{4b} + 100\sqrt{0,16b}$</p><p>Упростим каждое слагаемое, вынося множители из-под знака корня:</p><p>$25\sqrt{b} - 0,5\sqrt{4}\sqrt{b} + 100\sqrt{0,16}\sqrt{b} = 25\sqrt{b} - 0,5 \cdot 2\sqrt{b} + 100 \cdot 0,4\sqrt{b}$</p><p>$25\sqrt{b} - 1\sqrt{b} + 40\sqrt{b}$</p><p>Сложим коэффициенты при $\sqrt{b}$:</p><p>$(25 - 1 + 40)\sqrt{b} = 64\sqrt{b}$</p><p><strong>Ответ:</strong> $64\sqrt{b}$</p><p><strong>з)</strong> $\sqrt{98a} + 3\sqrt{242a} - 17\sqrt{512a}$</p><p>Упростим каждый корень, вынося из-под знака корня множители, являющиеся полными квадратами:</p><p>$\sqrt{49 \cdot 2a} + 3\sqrt{121 \cdot 2a} - 17\sqrt{256 \cdot 2a}$</p><p>$= \sqrt{49}\sqrt{2a} + 3\sqrt{121}\sqrt{2a} - 17\sqrt{256}\sqrt{2a}$</p><p>$= 7\sqrt{2a} + 3 \cdot 11\sqrt{2a} - 17 \cdot 16\sqrt{2a}$</p><p>$= 7\sqrt{2a} + 33\sqrt{2a} - 272\sqrt{2a}$</p><p>Приведем подобные слагаемые:</p><p>$(7 + 33 - 272)\sqrt{2a} = (40 - 272)\sqrt{2a} = -232\sqrt{2a}$</p><p><strong>Ответ:</strong> $-232\sqrt{2a}$</p>" ] #original: array:6 [ "id" => 1549294 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107} "task" => array:2 [ "refs" => "1113591" "type" => "task" ] "text" => "<p><strong>а)</strong> $\frac{3-2b}{a+b} \cdot \frac{9b+9a}{4b^2-9} - \frac{6b}{2b+3}$</p><p>Сначала упростим первое слагаемое. Разложим числитель второй дроби и знаменатель на множители: $9b+9a=9(a+b)$ и $4b^2-9=(2b-3)(2b+3)$. Также заметим, что $3-2b=-(2b-3)$.</p><p>$\frac{-(2b-3)}{a+b} \cdot \frac{9(a+b)}{(2b-3)(2b+3)} - \frac{6b}{2b+3}$</p><p>Сократим общие множители $(a+b)$ и $(2b-3)$:</p><p>$\frac{-1}{1} \cdot \frac{9}{2b+3} - \frac{6b}{2b+3} = -\frac{9}{2b+3} - \frac{6b}{2b+3}$</p><p>Теперь вычтем дроби с одинаковым знаменателем:</p><p>$\frac{-9-6b}{2b+3} = \frac{-3(3+2b)}{2b+3} = -3$</p><p><strong>Ответ:</strong> $-3$</p><p><strong>б)</strong> $\frac{16a^2 - 25c^2}{ac+c^2} : \frac{5c-4a}{3c} + \frac{13a+14c}{a+c}$</p><p>Выполним сначала деление. Для этого разложим на множители числитель и знаменатель первой дроби: $16a^2-25c^2=(4a-5c)(4a+5c)$ и $ac+c^2=c(a+c)$. Заметим, что $5c-4a=-(4a-5c)$. Деление заменяем на умножение на обратную дробь:</p><p>$\frac{(4a-5c)(4a+5c)}{c(a+c)} \cdot \frac{3c}{-(4a-5c)} + \frac{13a+14c}{a+c}$</p><p>Сократим общие множители $c$ и $(4a-5c)$:</p><p>$\frac{4a+5c}{a+c} \cdot \frac{3}{-1} + \frac{13a+14c}{a+c} = \frac{-3(4a+5c)}{a+c} + \frac{13a+14c}{a+c} = \frac{-12a-15c}{a+c} + \frac{13a+14c}{a+c}$</p><p>Сложим дроби с одинаковым знаменателем:</p><p>$\frac{-12a-15c+13a+14c}{a+c} = \frac{a-c}{a+c}$</p><p><strong>Ответ:</strong> $\frac{a-c}{a+c}$</p><p><strong>в)</strong> $(\frac{b+2}{b^2-3b+9} - \frac{6}{b^3+27}) : \frac{3b+15}{2b^2-6b+18}$</p><p>Упростим выражение в скобках. Используем формулу суммы кубов: $b^3+27=b^3+3^3=(b+3)(b^2-3b+9)$. Приведем дроби к общему знаменателю:</p><p>$\frac{(b+2)(b+3)}{(b+3)(b^2-3b+9)} - \frac{6}{(b+3)(b^2-3b+9)} = \frac{b^2+3b+2b+6-6}{(b+3)(b^2-3b+9)} = \frac{b^2+5b}{b^3+27} = \frac{b(b+5)}{b^3+27}$</p><p>Теперь выполним деление. Разложим на множители делитель: $3b+15=3(b+5)$ и $2b^2-6b+18=2(b^2-3b+9)$.</p><p>$\frac{b(b+5)}{b^3+27} : \frac{3(b+5)}{2(b^2-3b+9)} = \frac{b(b+5)}{(b+3)(b^2-3b+9)} \cdot \frac{2(b^2-3b+9)}{3(b+5)}$</p><p>Сократим общие множители $(b+5)$ и $(b^2-3b+9)$:</p><p>$\frac{b}{b+3} \cdot \frac{2}{3} = \frac{2b}{3(b+3)}$</p><p><strong>Ответ:</strong> $\frac{2b}{3(b+3)}$</p><p><strong>г)</strong> $(\frac{a-1}{a^2+2a+4} - \frac{2}{a^3-8}) : \frac{3-a}{a^2+2a+4}$</p><p>Упростим выражение в скобках, используя формулу разности кубов: $a^3-8=a^3-2^3=(a-2)(a^2+2a+4)$.</p><p>$\frac{(a-1)(a-2)}{(a-2)(a^2+2a+4)} - \frac{2}{(a-2)(a^2+2a+4)} = \frac{a^2-2a-a+2-2}{(a-2)(a^2+2a+4)} = \frac{a^2-3a}{a^3-8} = \frac{a(a-3)}{a^3-8}$</p><p>Теперь выполним деление. Заметим, что $3-a=-(a-3)$.</p><p>$\frac{a(a-3)}{a^3-8} : \frac{-(a-3)}{a^2+2a+4} = \frac{a(a-3)}{(a-2)(a^2+2a+4)} \cdot \frac{a^2+2a+4}{-(a-3)}$</p><p>Сократим общие множители $(a-3)$ и $(a^2+2a+4)$:</p><p>$\frac{a}{a-2} \cdot \frac{1}{-1} = -\frac{a}{a-2}$</p><p><strong>Ответ:</strong> $-\frac{a}{a-2}$</p><p><strong>д)</strong> $\frac{2a+1}{a-7} + (\frac{a}{a+8} + \frac{a}{7-a}) : \frac{a}{8+a}$</p><p>Сначала выполним действия в скобках: $7-a=-(a-7)$.</p><p>$\frac{a}{a+8} + \frac{a}{-(a-7)} = \frac{a}{a+8} - \frac{a}{a-7} = \frac{a(a-7)-a(a+8)}{(a+8)(a-7)} = \frac{a^2-7a-a^2-8a}{(a+8)(a-7)} = \frac{-15a}{(a+8)(a-7)}$</p><p>Теперь выполним деление:</p><p>$\frac{-15a}{(a+8)(a-7)} : \frac{a}{a+8} = \frac{-15a}{(a+8)(a-7)} \cdot \frac{a+8}{a}$</p><p>Сократим общие множители $a$ и $(a+8)$: $\frac{-15}{a-7}$.</p><p>Теперь выполним сложение:</p><p>$\frac{2a+1}{a-7} + \frac{-15}{a-7} = \frac{2a+1-15}{a-7} = \frac{2a-14}{a-7} = \frac{2(a-7)}{a-7} = 2$</p><p><strong>Ответ:</strong> $2$</p><p><strong>е)</strong> $(\frac{b}{2b-3} - \frac{b}{3+2b}) : \frac{b}{9+12b+4b^2} + \frac{45-6b}{3-2b}$</p><p>Упростим выражение в скобках:</p><p>$\frac{b}{2b-3} - \frac{b}{2b+3} = \frac{b(2b+3)-b(2b-3)}{(2b-3)(2b+3)} = \frac{2b^2+3b-2b^2+3b}{4b^2-9} = \frac{6b}{4b^2-9}$</p><p>Теперь деление. Знаменатель делителя $9+12b+4b^2 = (3+2b)^2$.</p><p>$\frac{6b}{4b^2-9} : \frac{b}{(2b+3)^2} = \frac{6b}{(2b-3)(2b+3)} \cdot \frac{(2b+3)^2}{b}$</p><p>Сокращаем $b$ и $(2b+3)$: $\frac{6(2b+3)}{2b-3} = \frac{12b+18}{2b-3}$.</p><p>Теперь сложение. Заметим, что $3-2b=-(2b-3)$.</p><p>$\frac{12b+18}{2b-3} + \frac{45-6b}{-(2b-3)} = \frac{12b+18}{2b-3} - \frac{45-6b}{2b-3} = \frac{12b+18-(45-6b)}{2b-3} = \frac{12b+18-45+6b}{2b-3} = \frac{18b-27}{2b-3} = \frac{9(2b-3)}{2b-3} = 9$</p><p><strong>Ответ:</strong> $9$</p><p><strong>ж)</strong> $25\sqrt{b} - 0,5\sqrt{4b} + 100\sqrt{0,16b}$</p><p>Упростим каждое слагаемое, вынося множители из-под знака корня:</p><p>$25\sqrt{b} - 0,5\sqrt{4}\sqrt{b} + 100\sqrt{0,16}\sqrt{b} = 25\sqrt{b} - 0,5 \cdot 2\sqrt{b} + 100 \cdot 0,4\sqrt{b}$</p><p>$25\sqrt{b} - 1\sqrt{b} + 40\sqrt{b}$</p><p>Сложим коэффициенты при $\sqrt{b}$:</p><p>$(25 - 1 + 40)\sqrt{b} = 64\sqrt{b}$</p><p><strong>Ответ:</strong> $64\sqrt{b}$</p><p><strong>з)</strong> $\sqrt{98a} + 3\sqrt{242a} - 17\sqrt{512a}$</p><p>Упростим каждый корень, вынося из-под знака корня множители, являющиеся полными квадратами:</p><p>$\sqrt{49 \cdot 2a} + 3\sqrt{121 \cdot 2a} - 17\sqrt{256 \cdot 2a}$</p><p>$= \sqrt{49}\sqrt{2a} + 3\sqrt{121}\sqrt{2a} - 17\sqrt{256}\sqrt{2a}$</p><p>$= 7\sqrt{2a} + 3 \cdot 11\sqrt{2a} - 17 \cdot 16\sqrt{2a}$</p><p>$= 7\sqrt{2a} + 33\sqrt{2a} - 272\sqrt{2a}$</p><p>Приведем подобные слагаемые:</p><p>$(7 + 33 - 272)\sqrt{2a} = (40 - 272)\sqrt{2a} = -232\sqrt{2a}$</p><p><strong>Ответ:</strong> $-232\sqrt{2a}$</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113592" "type" => "task" ] "previous" => array:2 [ "refs" => null "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1079 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1076 #items: array:1 [ 0 => App\Models\BookPage {#1040 #connection: "mysql" 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#items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1117 #items: array:1 [ 0 => App\Models\Element {#1116 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1261766 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1118 #items: array:1 [ 0 => App\Models\Edition {#1083} ] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1098341" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "3192" "height" => 4602 "path" => "/media/algebra_10/Abylkasymova-u/0-1/4-1.webp?ts=1752147445" ] ] ] #original: array:6 [ "id" => 1261766 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1118} "book_page" => array:2 [ "refs" => "1098341" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "3192" "height" => 4602 "path" => "/media/algebra_10/Abylkasymova-u/0-1/4-1.webp?ts=1752147445" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1098342" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1098340" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1176 #items: array:5 [ 0 => App\Models\Task {#1188 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113591 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "4" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-1" "field_display_title" => "1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1189 #items: array:1 [ 0 => App\Models\Term {#1036} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1190 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1191 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1192 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1193 #items: array:3 [ 0 => App\Models\Element {#1082} 1 => App\Models\Element {#1089} 2 => App\Models\Element {#1097} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113592" "type" => "task" ] "previous" => array:2 [ "refs" => null "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1194 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098341" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113591 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "4" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-1" "field_display_title" => "1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1189} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1190} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1191} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1192} "content" => Illuminate\Database\Eloquent\Collection {#1193} "next" => array:2 [ "refs" => "1113592" "type" => "task" ] "previous" => array:2 [ "refs" => null "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1194} "page" => array:2 [ "refs" => "1098341" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Task {#1195 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113592 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "4" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-2" "field_display_title" => "2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1196 #items: array:1 [ 0 => App\Models\Term {#1036} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1197 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1198 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1199 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1202 #items: array:3 [ 0 => App\Models\Element {#1211 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ …7] #original: array:7 [ …7] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Element {#1213 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Element {#1215 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113593" "type" => "task" ] "previous" => array:2 [ "refs" => "1113591" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1208 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098341" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113592 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "4" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-2" "field_display_title" => "2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1196} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1197} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1198} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1199} "content" => Illuminate\Database\Eloquent\Collection {#1202} "next" => array:2 [ "refs" => "1113593" "type" => "task" ] "previous" => array:2 [ "refs" => "1113591" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1208} "page" => array:2 [ "refs" => "1098341" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Task {#1201 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113593 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "4" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-3" "field_display_title" => "3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1200 #items: array:1 [ 0 => App\Models\Term {#1036} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1207 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1209 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1205 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1204 #items: array:3 [ 0 => App\Models\Element {#1225 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ …7] #original: array:7 [ …7] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Element {#1227 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Element {#1229 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113594" "type" => "task" ] "previous" => array:2 [ "refs" => "1113592" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1222 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098341" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113593 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "4" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-3" "field_display_title" => "3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1200} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1207} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1209} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1205} "content" => Illuminate\Database\Eloquent\Collection {#1204} "next" => array:2 [ "refs" => "1113594" "type" => "task" ] "previous" => array:2 [ "refs" => "1113592" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1222} "page" => array:2 [ "refs" => "1098341" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Task {#1206 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113594 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "4" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-4" "field_display_title" => "4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1203 #items: array:1 [ 0 => App\Models\Term {#1036} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1221 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1223 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1219 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1218 #items: array:3 [ 0 => App\Models\Element {#1239 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ …7] #original: array:7 [ …7] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Element {#1241 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Element {#1243 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113595" "type" => "task" ] "previous" => array:2 [ "refs" => "1113593" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1236 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098341" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113594 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "4" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-4" "field_display_title" => "4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1203} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1221} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1223} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1219} "content" => Illuminate\Database\Eloquent\Collection {#1218} "next" => array:2 [ "refs" => "1113595" "type" => "task" ] "previous" => array:2 [ "refs" => "1113593" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1236} "page" => array:2 [ "refs" => "1098341" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Task {#1220 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113595 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "4" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-5" "field_display_title" => "5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1217 #items: array:1 [ 0 => App\Models\Term {#1036} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1235 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1237 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1233 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1232 #items: array:3 [ 0 => App\Models\Element {#1253 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ …7] #original: array:7 [ …7] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Element {#1255 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Element {#1257 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113596" "type" => "task" ] "previous" => array:2 [ "refs" => "1113594" "type" => "task" ] 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№1 (с. 4)
Условие. №1 (с. 4)
скриншот условия
1. Упростите выражение:
а) $\frac{3-2b}{a+b} : \frac{9b+9a}{4b^2-9} - \frac{6b}{2b+3}$;
б) $\frac{16a^2-25c^2}{ac+c^2} : \frac{5c-4a}{3c} + \frac{13a+14c}{a+c}$;
В) $(\frac{b+2}{b^2-3b+9} - \frac{6}{b^3+27}) : \frac{3b+15}{2b^2-6b+18}$;
Г) $(\frac{a-1}{a^2+2a+4} - \frac{2}{a^3-8}) : \frac{3-a}{a^2+2a+4}$;
Д) $\frac{2a+1}{a-7} + (\frac{a}{a+8} + \frac{a}{7-a}) : \frac{a}{8+a}$;
е) $(\frac{b}{2b-3} - \frac{b}{3+2b}) : \frac{b}{9+12b+4b^2} + \frac{45-6b}{3-2b}$;
ж) $25\sqrt{b} - 0.5\sqrt{4b} + 100\sqrt{0.16b}$;
3) $\sqrt{98a} + 3\sqrt{242a} - 17\sqrt{512a}$.
Решение. №1 (с. 4)
Решение 2. №1 (с. 4)
а) $\frac{3-2b}{a+b} \cdot \frac{9b+9a}{4b^2-9} - \frac{6b}{2b+3}$
Сначала упростим первое слагаемое. Разложим числитель второй дроби и знаменатель на множители: $9b+9a=9(a+b)$ и $4b^2-9=(2b-3)(2b+3)$. Также заметим, что $3-2b=-(2b-3)$.
$\frac{-(2b-3)}{a+b} \cdot \frac{9(a+b)}{(2b-3)(2b+3)} - \frac{6b}{2b+3}$
Сократим общие множители $(a+b)$ и $(2b-3)$:
$\frac{-1}{1} \cdot \frac{9}{2b+3} - \frac{6b}{2b+3} = -\frac{9}{2b+3} - \frac{6b}{2b+3}$
Теперь вычтем дроби с одинаковым знаменателем:
$\frac{-9-6b}{2b+3} = \frac{-3(3+2b)}{2b+3} = -3$
Ответ: $-3$
б) $\frac{16a^2 - 25c^2}{ac+c^2} : \frac{5c-4a}{3c} + \frac{13a+14c}{a+c}$
Выполним сначала деление. Для этого разложим на множители числитель и знаменатель первой дроби: $16a^2-25c^2=(4a-5c)(4a+5c)$ и $ac+c^2=c(a+c)$. Заметим, что $5c-4a=-(4a-5c)$. Деление заменяем на умножение на обратную дробь:
$\frac{(4a-5c)(4a+5c)}{c(a+c)} \cdot \frac{3c}{-(4a-5c)} + \frac{13a+14c}{a+c}$
Сократим общие множители $c$ и $(4a-5c)$:
$\frac{4a+5c}{a+c} \cdot \frac{3}{-1} + \frac{13a+14c}{a+c} = \frac{-3(4a+5c)}{a+c} + \frac{13a+14c}{a+c} = \frac{-12a-15c}{a+c} + \frac{13a+14c}{a+c}$
Сложим дроби с одинаковым знаменателем:
$\frac{-12a-15c+13a+14c}{a+c} = \frac{a-c}{a+c}$
Ответ: $\frac{a-c}{a+c}$
в) $(\frac{b+2}{b^2-3b+9} - \frac{6}{b^3+27}) : \frac{3b+15}{2b^2-6b+18}$
Упростим выражение в скобках. Используем формулу суммы кубов: $b^3+27=b^3+3^3=(b+3)(b^2-3b+9)$. Приведем дроби к общему знаменателю:
$\frac{(b+2)(b+3)}{(b+3)(b^2-3b+9)} - \frac{6}{(b+3)(b^2-3b+9)} = \frac{b^2+3b+2b+6-6}{(b+3)(b^2-3b+9)} = \frac{b^2+5b}{b^3+27} = \frac{b(b+5)}{b^3+27}$
Теперь выполним деление. Разложим на множители делитель: $3b+15=3(b+5)$ и $2b^2-6b+18=2(b^2-3b+9)$.
$\frac{b(b+5)}{b^3+27} : \frac{3(b+5)}{2(b^2-3b+9)} = \frac{b(b+5)}{(b+3)(b^2-3b+9)} \cdot \frac{2(b^2-3b+9)}{3(b+5)}$
Сократим общие множители $(b+5)$ и $(b^2-3b+9)$:
$\frac{b}{b+3} \cdot \frac{2}{3} = \frac{2b}{3(b+3)}$
Ответ: $\frac{2b}{3(b+3)}$
г) $(\frac{a-1}{a^2+2a+4} - \frac{2}{a^3-8}) : \frac{3-a}{a^2+2a+4}$
Упростим выражение в скобках, используя формулу разности кубов: $a^3-8=a^3-2^3=(a-2)(a^2+2a+4)$.
$\frac{(a-1)(a-2)}{(a-2)(a^2+2a+4)} - \frac{2}{(a-2)(a^2+2a+4)} = \frac{a^2-2a-a+2-2}{(a-2)(a^2+2a+4)} = \frac{a^2-3a}{a^3-8} = \frac{a(a-3)}{a^3-8}$
Теперь выполним деление. Заметим, что $3-a=-(a-3)$.
$\frac{a(a-3)}{a^3-8} : \frac{-(a-3)}{a^2+2a+4} = \frac{a(a-3)}{(a-2)(a^2+2a+4)} \cdot \frac{a^2+2a+4}{-(a-3)}$
Сократим общие множители $(a-3)$ и $(a^2+2a+4)$:
$\frac{a}{a-2} \cdot \frac{1}{-1} = -\frac{a}{a-2}$
Ответ: $-\frac{a}{a-2}$
д) $\frac{2a+1}{a-7} + (\frac{a}{a+8} + \frac{a}{7-a}) : \frac{a}{8+a}$
Сначала выполним действия в скобках: $7-a=-(a-7)$.
$\frac{a}{a+8} + \frac{a}{-(a-7)} = \frac{a}{a+8} - \frac{a}{a-7} = \frac{a(a-7)-a(a+8)}{(a+8)(a-7)} = \frac{a^2-7a-a^2-8a}{(a+8)(a-7)} = \frac{-15a}{(a+8)(a-7)}$
Теперь выполним деление:
$\frac{-15a}{(a+8)(a-7)} : \frac{a}{a+8} = \frac{-15a}{(a+8)(a-7)} \cdot \frac{a+8}{a}$
Сократим общие множители $a$ и $(a+8)$: $\frac{-15}{a-7}$.
Теперь выполним сложение:
$\frac{2a+1}{a-7} + \frac{-15}{a-7} = \frac{2a+1-15}{a-7} = \frac{2a-14}{a-7} = \frac{2(a-7)}{a-7} = 2$
Ответ: $2$
е) $(\frac{b}{2b-3} - \frac{b}{3+2b}) : \frac{b}{9+12b+4b^2} + \frac{45-6b}{3-2b}$
Упростим выражение в скобках:
$\frac{b}{2b-3} - \frac{b}{2b+3} = \frac{b(2b+3)-b(2b-3)}{(2b-3)(2b+3)} = \frac{2b^2+3b-2b^2+3b}{4b^2-9} = \frac{6b}{4b^2-9}$
Теперь деление. Знаменатель делителя $9+12b+4b^2 = (3+2b)^2$.
$\frac{6b}{4b^2-9} : \frac{b}{(2b+3)^2} = \frac{6b}{(2b-3)(2b+3)} \cdot \frac{(2b+3)^2}{b}$
Сокращаем $b$ и $(2b+3)$: $\frac{6(2b+3)}{2b-3} = \frac{12b+18}{2b-3}$.
Теперь сложение. Заметим, что $3-2b=-(2b-3)$.
$\frac{12b+18}{2b-3} + \frac{45-6b}{-(2b-3)} = \frac{12b+18}{2b-3} - \frac{45-6b}{2b-3} = \frac{12b+18-(45-6b)}{2b-3} = \frac{12b+18-45+6b}{2b-3} = \frac{18b-27}{2b-3} = \frac{9(2b-3)}{2b-3} = 9$
Ответ: $9$
ж) $25\sqrt{b} - 0,5\sqrt{4b} + 100\sqrt{0,16b}$
Упростим каждое слагаемое, вынося множители из-под знака корня:
$25\sqrt{b} - 0,5\sqrt{4}\sqrt{b} + 100\sqrt{0,16}\sqrt{b} = 25\sqrt{b} - 0,5 \cdot 2\sqrt{b} + 100 \cdot 0,4\sqrt{b}$
$25\sqrt{b} - 1\sqrt{b} + 40\sqrt{b}$
Сложим коэффициенты при $\sqrt{b}$:
$(25 - 1 + 40)\sqrt{b} = 64\sqrt{b}$
Ответ: $64\sqrt{b}$
з) $\sqrt{98a} + 3\sqrt{242a} - 17\sqrt{512a}$
Упростим каждый корень, вынося из-под знака корня множители, являющиеся полными квадратами:
$\sqrt{49 \cdot 2a} + 3\sqrt{121 \cdot 2a} - 17\sqrt{256 \cdot 2a}$
$= \sqrt{49}\sqrt{2a} + 3\sqrt{121}\sqrt{2a} - 17\sqrt{256}\sqrt{2a}$
$= 7\sqrt{2a} + 3 \cdot 11\sqrt{2a} - 17 \cdot 16\sqrt{2a}$
$= 7\sqrt{2a} + 33\sqrt{2a} - 272\sqrt{2a}$
Приведем подобные слагаемые:
$(7 + 33 - 272)\sqrt{2a} = (40 - 272)\sqrt{2a} = -232\sqrt{2a}$
Ответ: $-232\sqrt{2a}$
Другие задания:
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