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Номер 9, страница 64 - гдз по алгебре 11 класс учебник Абылкасымова, Жумагулова
Авторы: Абылкасымова А. Е., Жумагулова З. А.
Тип: Учебник
Издательство: Мектеп
Год издания: 2019 - 2026
Популярные ГДЗ в 11 классе
Глава II. Степени и корни. Степенная функция. Проверь себя! - номер 9, страница 64.
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" => 1125006 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "64" "field_page_end" => null "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019/prceb64-9" "field_display_title" => "9" "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" => 1124770 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Степени и корни. Степенная функция" "field_branch_order" => "II" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1050 #items: array:1 [ 0 => App\Models\Term {#1047 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => 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_page_start" => "39" "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 {#1058 #items: array:1 [ 0 => App\Models\Book {#1057 #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" => 4297 "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 {#1062 #items: array:1 [ 0 => App\Models\Term {#1059 #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 {#1064 #items: array:1 [ 0 => App\Models\Term {#1061 #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" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ "field_accusative_case" => "одиннадцатый" "field_creative_case" => "одиннадцатым" "field_dative_case" => "одиннадцатому" "field_genitive_case" => "одинадцатого" "field_nominative_case" => "одинадцатый" "field_prepositional_case" => "одиннадцатом" ] "field_translit" => "odinadcatyj" ] #original: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ "field_accusative_case" => "одиннадцатый" "field_creative_case" => "одиннадцатым" "field_dative_case" => "одиннадцатому" "field_genitive_case" => "одинадцатого" "field_nominative_case" => "одинадцатый" "field_prepositional_case" => "одиннадцатом" ] "field_translit" => "odinadcatyj" ] #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 {#1066 #items: array:1 [ 0 => App\Models\Term {#1063 #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 {#1060 #items: array:2 [ 0 => App\Models\Term {#1075 #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 {#1076 #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 {#1072 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1070 #items: array:1 [ 0 => App\Models\Term {#1067 #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 {#1081 #items: array:1 [ 0 => App\Models\Term {#1080 #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 {#1084 #items: array:1 [ 0 => App\Models\Term {#1083 #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 {#1068 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1074 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1065 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1078 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1077 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1069 #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" => "Общественно-гуманитарного направления" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "530" "field_priority" => "2" "field_default_folder" => "/algebra_11/abylkasymova1-u19/" "field_isbn" => null "field_cover" => array:1 [ 0 => "/media/algebra_11/abylkasymova1-u19/covers/cover1.webp?ts=1752223853" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_11/abylkasymova1-u19/covers/cover1.webp?ts=1752223853" "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 {#1085 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1086 #items: [] #escapeWhenCastingToString: false } "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1087 #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" => 382 "subject" => 542 "class_subject" => 388 ] ] #original: array:50 [ "id" => 4297 "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 {#1062} "field_class" => Illuminate\Database\Eloquent\Collection {#1064} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1066} "field_author" => Illuminate\Database\Eloquent\Collection {#1060} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1072} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1070} "field_country" => Illuminate\Database\Eloquent\Collection {#1081} "field_city" => Illuminate\Database\Eloquent\Collection {#1084} "field_series" => Illuminate\Database\Eloquent\Collection {#1068} "field_umk" => Illuminate\Database\Eloquent\Collection {#1074} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1065} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1078} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1077} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1069} "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" => "Общественно-гуманитарного направления" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "530" "field_priority" => "2" "field_default_folder" => "/algebra_11/abylkasymova1-u19/" "field_isbn" => null "field_cover" => array:1 [ 0 => "/media/algebra_11/abylkasymova1-u19/covers/cover1.webp?ts=1752223853" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_11/abylkasymova1-u19/covers/cover1.webp?ts=1752223853" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1071} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1085} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1086} "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1087} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 382 "subject" => 542 "class_subject" => 388 ] ] #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 {#1048 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1124770 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Степени и корни. Степенная функция" "field_branch_order" => "II" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1050} "field_page_start" => "39" "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 {#1058} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1048} ] #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 {#1109 #items: array:2 [ 0 => App\Models\Branch {#1117 #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" => 1124770 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Степени и корни. Степенная функция" "field_branch_order" => "II" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1118 #items: array:1 [ 0 => App\Models\Term {#1047} ] #escapeWhenCastingToString: false } "field_page_start" => "39" "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 {#1119 #items: array:1 [ 0 => App\Models\Book {#1057} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1120 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1124770 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Степени и корни. Степенная функция" "field_branch_order" => "II" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1118} "field_page_start" => "39" "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 {#1119} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1120} ] #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\Branch {#1121 #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" => 1124776 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Проверь себя!" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1122 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "64" "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 {#1123 #items: array:1 [ 0 => App\Models\Book {#1057} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1124 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1124776 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Проверь себя!" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1122} "field_page_start" => "64" "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 {#1123} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1124} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1038 #items: array:2 [ 0 => App\Models\Element {#1100 #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" => 1308516 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1091 #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" => 4970 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1089 #items: array:1 [ …1] #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 {#1051 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1088 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1053 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4970 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1089} "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 {#1051} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1088} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1053} "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" => "1125006" "type" => "task" ] "text" => "<p><strong>9. Сократите дробь</strong> $\frac{a - b}{a^{0.5}b^{0.5} - b}$ :</p><p><strong>A</strong> $\frac{a^{0.5} - b^{0.5}}{b^{0.5}}$ ; B) $\frac{a^{0.5} + b^{0.5}}{b^{0.5}}$ ; C) $\frac{a^{0.5}}{b}$ ; D) $\frac{a^{0.5} - b^{0.5}}{b}$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "9-1.jpg" "alt" => null "width" => "1547" "height" => 291 "path" => "/media/algebra_11/abylkasymova1-u19/0-prceb64/9-1.webp?ts=1753819482" ] ] ] #original: array:7 [ "id" => 1308516 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1091} "task" => array:2 [ "refs" => "1125006" "type" => "task" ] "text" => "<p><strong>9. Сократите дробь</strong> $\frac{a - b}{a^{0.5}b^{0.5} - b}$ :</p><p><strong>A</strong> $\frac{a^{0.5} - b^{0.5}}{b^{0.5}}$ ; B) $\frac{a^{0.5} + b^{0.5}}{b^{0.5}}$ ; C) $\frac{a^{0.5}}{b}$ ; D) $\frac{a^{0.5} - b^{0.5}}{b}$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "9-1.jpg" "alt" => null "width" => "1547" "height" => 291 "path" => "/media/algebra_11/abylkasymova1-u19/0-prceb64/9-1.webp?ts=1753819482" ] ] ] #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 {#1098 #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" => 1553542 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1052 #items: array:1 [ 0 => App\Models\Edition {#1049 #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" => 5663 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1108 #items: 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 {#1110 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1041 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1107 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5663 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1108} "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 {#1110} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1041} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1107} "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" => "1125006" "type" => "task" ] "text" => "<p>Для того чтобы сократить заданную дробь $\frac{a - b}{a^{0.5}b^{0.5} - b}$, необходимо разложить на множители её числитель и знаменатель.</p><p><strong>1. Преобразование числителя.</strong></p><p>Числитель $a - b$ можно рассматривать как разность квадратов, так как $a = (a^{0.5})^2$ и $b = (b^{0.5})^2$.</p><p>Применим формулу разности квадратов $x^2 - y^2 = (x-y)(x+y)$:</p><p>$a - b = (a^{0.5})^2 - (b^{0.5})^2 = (a^{0.5} - b^{0.5})(a^{0.5} + b^{0.5})$.</p><p><strong>2. Преобразование знаменателя.</strong></p><p>В знаменателе $a^{0.5}b^{0.5} - b$ вынесем общий множитель за скобки. Для этого представим $b$ как $b^{0.5} \cdot b^{0.5}$.</p><p>$a^{0.5}b^{0.5} - b = a^{0.5}b^{0.5} - b^{0.5}b^{0.5}$.</p><p>Общий множитель здесь — $b^{0.5}$. Выносим его:</p><p>$b^{0.5}(a^{0.5} - b^{0.5})$.</p><p><strong>3. Сокращение дроби.</strong></p><p>Теперь подставим полученные выражения обратно в дробь:</p><p>$\frac{(a^{0.5} - b^{0.5})(a^{0.5} + b^{0.5})}{b^{0.5}(a^{0.5} - b^{0.5})}$.</p><p>Мы видим, что в числителе и знаменателе есть одинаковый множитель $(a^{0.5} - b^{0.5})$, который можно сократить (при условии, что он не равен нулю, т.е. $a \neq b$):</p><p>$\frac{\cancel{(a^{0.5} - b^{0.5})}(a^{0.5} + b^{0.5})}{b^{0.5}\cancel{(a^{0.5} - b^{0.5})}} = \frac{a^{0.5} + b^{0.5}}{b^{0.5}}$.</p><p>Полученное выражение соответствует варианту ответа B).</p><p><strong>Ответ:</strong> B) $\frac{a^{0.5} + b^{0.5}}{b^{0.5}}$</p>" ] #original: array:6 [ "id" => 1553542 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1052} "task" => array:2 [ "refs" => "1125006" "type" => "task" ] "text" => "<p>Для того чтобы сократить заданную дробь $\frac{a - b}{a^{0.5}b^{0.5} - b}$, необходимо разложить на множители её числитель и знаменатель.</p><p><strong>1. Преобразование числителя.</strong></p><p>Числитель $a - b$ можно рассматривать как разность квадратов, так как $a = (a^{0.5})^2$ и $b = (b^{0.5})^2$.</p><p>Применим формулу разности квадратов $x^2 - y^2 = (x-y)(x+y)$:</p><p>$a - b = (a^{0.5})^2 - (b^{0.5})^2 = (a^{0.5} - b^{0.5})(a^{0.5} + b^{0.5})$.</p><p><strong>2. Преобразование знаменателя.</strong></p><p>В знаменателе $a^{0.5}b^{0.5} - b$ вынесем общий множитель за скобки. Для этого представим $b$ как $b^{0.5} \cdot b^{0.5}$.</p><p>$a^{0.5}b^{0.5} - b = a^{0.5}b^{0.5} - b^{0.5}b^{0.5}$.</p><p>Общий множитель здесь — $b^{0.5}$. Выносим его:</p><p>$b^{0.5}(a^{0.5} - b^{0.5})$.</p><p><strong>3. Сокращение дроби.</strong></p><p>Теперь подставим полученные выражения обратно в дробь:</p><p>$\frac{(a^{0.5} - b^{0.5})(a^{0.5} + b^{0.5})}{b^{0.5}(a^{0.5} - b^{0.5})}$.</p><p>Мы видим, что в числителе и знаменателе есть одинаковый множитель $(a^{0.5} - b^{0.5})$, который можно сократить (при условии, что он не равен нулю, т.е. $a \neq b$):</p><p>$\frac{\cancel{(a^{0.5} - b^{0.5})}(a^{0.5} + b^{0.5})}{b^{0.5}\cancel{(a^{0.5} - b^{0.5})}} = \frac{a^{0.5} + b^{0.5}}{b^{0.5}}$.</p><p>Полученное выражение соответствует варианту ответа B).</p><p><strong>Ответ:</strong> B) $\frac{a^{0.5} + b^{0.5}}{b^{0.5}}$</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" => "1125007" "type" => "task" ] "previous" => array:2 [ "refs" => "1125005" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1103 #items: array:1 [ 0 => App\Models\Book {#1057} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1040 #items: array:1 [ 0 => App\Models\BookPage {#1104 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 1100042 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "64" "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019/page-64" "field_display_title" => "64" "field_folder" => "1" "field_image_name" => "64" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1099 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1095 #items: array:1 [ 0 => App\Models\Book {#1057} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1097 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1125 #items: array:1 [ 0 => App\Models\Element {#1111 #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" => 1263466 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1127 #items: array:1 [ …1] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1100042" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #original: array:6 [ "id" => 1263466 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1127} "book_page" => array:2 [ "refs" => "1100042" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #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" => "1100043" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1100041" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1186 #items: array:10 [ 0 => App\Models\Task {#1202 #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" => 1124998 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "64" "field_page_end" => null "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019/prceb64-1" "field_display_title" => "1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1203 #items: array:1 [ …1] #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 {#1204 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1205 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1206 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1208 #items: array:2 [ …2] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1124999" "type" => "task" ] "previous" => array:2 [ "refs" => "1124997" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1214 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1100042" "type" => "book_page" ] ] #original: array:24 [ "id" => 1124998 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "64" "field_page_end" => null "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019/prceb64-1" "field_display_title" => "1" "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 {#1204} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1205} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1206} "content" => Illuminate\Database\Eloquent\Collection {#1208} "next" => array:2 [ "refs" => "1124999" "type" => "task" ] "previous" => array:2 [ "refs" => "1124997" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1214} "page" => array:2 [ "refs" => "1100042" "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 {#1209 #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" => 1124999 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "64" "field_page_end" => null "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019/prceb64-2" "field_display_title" => "2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1207 #items: array:1 [ …1] #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 {#1216 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1212 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1210 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1211 #items: array:2 [ …2] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1125000" "type" => "task" ] "previous" => array:2 [ "refs" => "1124998" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1226 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1100042" "type" => "book_page" ] ] #original: array:24 [ "id" => 1124999 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "64" "field_page_end" => null "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019/prceb64-2" "field_display_title" => "2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1207} "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 {#1216} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1212} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1210} "content" => Illuminate\Database\Eloquent\Collection {#1211} "next" => array:2 [ "refs" => "1125000" "type" => "task" ] "previous" => array:2 [ "refs" => "1124998" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1226} "page" => array:2 [ "refs" => "1100042" "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 {#1221 #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" => 1125000 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "64" "field_page_end" => null "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019/prceb64-3" "field_display_title" => "3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1213 #items: array:1 [ …1] #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 {#1228 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1224 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1222 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1223 #items: array:2 [ …2] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1125001" "type" => "task" ] "previous" => array:2 [ "refs" => "1124999" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1238 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1100042" "type" => "book_page" ] ] #original: array:24 [ "id" => 1125000 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "64" "field_page_end" => null "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019/prceb64-3" "field_display_title" => "3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1213} "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 {#1228} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1224} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1222} "content" => Illuminate\Database\Eloquent\Collection {#1223} "next" => array:2 [ "refs" => "1125001" "type" => "task" ] "previous" => array:2 [ "refs" => "1124999" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1238} "page" => array:2 [ "refs" => "1100042" "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 {#1233 #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" => 1125001 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "64" "field_page_end" => null "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019/prceb64-4" "field_display_title" => "4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1225 #items: array:1 [ …1] #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 {#1240 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1236 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1234 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1235 #items: array:2 [ …2] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1125002" "type" => "task" ] "previous" => array:2 [ "refs" => "1125000" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1250 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1100042" "type" => "book_page" ] ] #original: array:24 [ "id" => 1125001 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "64" "field_page_end" => null "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019/prceb64-4" "field_display_title" => "4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1225} "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 {#1240} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1236} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1234} "content" => Illuminate\Database\Eloquent\Collection {#1235} "next" => array:2 [ "refs" => "1125002" "type" => "task" ] "previous" => array:2 [ "refs" => "1125000" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1250} "page" => array:2 [ "refs" => "1100042" "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 {#1245 #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" => 1125002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "64" "field_page_end" => null "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019/prceb64-5" "field_display_title" => "5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1237 #items: array:1 [ …1] #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 {#1252 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1248 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1246 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1247 #items: array:2 [ …2] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1125003" "type" => "task" ] "previous" => array:2 [ "refs" => "1125001" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1262 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1100042" "type" => "book_page" ] ] #original: array:24 [ "id" => 1125002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "64" "field_page_end" => null "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019/prceb64-5" "field_display_title" => "5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1237} "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 {#1252} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1248} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1246} "content" => Illuminate\Database\Eloquent\Collection {#1247} "next" => array:2 [ "refs" => "1125003" "type" => "task" ] "previous" => array:2 [ "refs" => "1125001" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1262} "page" => array:2 [ "refs" => "1100042" "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 => "*" ] } 5 => App\Models\Task {#1257 #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" => 1125003 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "64" "field_page_end" => null "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019/prceb64-6" "field_display_title" => "6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1249 #items: array:1 [ …1] #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 {#1264 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1260 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1258 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1259 #items: array:2 [ …2] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1125004" "type" => "task" ] "previous" => array:2 [ "refs" => "1125002" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1274 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1100042" "type" => "book_page" ] ] #original: array:24 [ "id" => 1125003 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "64" "field_page_end" => null "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019/prceb64-6" "field_display_title" => "6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1249} "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 {#1264} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1260} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1258} "content" => Illuminate\Database\Eloquent\Collection {#1259} "next" => array:2 [ "refs" => "1125004" "type" => "task" ] "previous" => array:2 [ "refs" => "1125002" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1274} "page" => array:2 [ "refs" => "1100042" "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 => "*" ] } 6 => App\Models\Task {#1269 #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" => 1125004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "64" "field_page_end" => null "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019/prceb64-7" "field_display_title" => "7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1261 #items: array:1 [ …1] #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 {#1276 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[] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1125006 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "64" "field_page_end" => null "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019/prceb64-9" "field_display_title" => "9" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037} "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} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1046} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1109} "content" => Illuminate\Database\Eloquent\Collection {#1038} "next" => array:2 [ "refs" => "1125007" "type" => "task" ] "previous" => 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№9 (с. 64)
Условие. №9 (с. 64)
скриншот условия
9. Сократите дробь $\frac{a - b}{a^{0.5}b^{0.5} - b}$ :
A $\frac{a^{0.5} - b^{0.5}}{b^{0.5}}$ ; B) $\frac{a^{0.5} + b^{0.5}}{b^{0.5}}$ ; C) $\frac{a^{0.5}}{b}$ ; D) $\frac{a^{0.5} - b^{0.5}}{b}$.
Решение 2 (rus). №9 (с. 64)
Для того чтобы сократить заданную дробь $\frac{a - b}{a^{0.5}b^{0.5} - b}$, необходимо разложить на множители её числитель и знаменатель.
1. Преобразование числителя.
Числитель $a - b$ можно рассматривать как разность квадратов, так как $a = (a^{0.5})^2$ и $b = (b^{0.5})^2$.
Применим формулу разности квадратов $x^2 - y^2 = (x-y)(x+y)$:
$a - b = (a^{0.5})^2 - (b^{0.5})^2 = (a^{0.5} - b^{0.5})(a^{0.5} + b^{0.5})$.
2. Преобразование знаменателя.
В знаменателе $a^{0.5}b^{0.5} - b$ вынесем общий множитель за скобки. Для этого представим $b$ как $b^{0.5} \cdot b^{0.5}$.
$a^{0.5}b^{0.5} - b = a^{0.5}b^{0.5} - b^{0.5}b^{0.5}$.
Общий множитель здесь — $b^{0.5}$. Выносим его:
$b^{0.5}(a^{0.5} - b^{0.5})$.
3. Сокращение дроби.
Теперь подставим полученные выражения обратно в дробь:
$\frac{(a^{0.5} - b^{0.5})(a^{0.5} + b^{0.5})}{b^{0.5}(a^{0.5} - b^{0.5})}$.
Мы видим, что в числителе и знаменателе есть одинаковый множитель $(a^{0.5} - b^{0.5})$, который можно сократить (при условии, что он не равен нулю, т.е. $a \neq b$):
$\frac{\cancel{(a^{0.5} - b^{0.5})}(a^{0.5} + b^{0.5})}{b^{0.5}\cancel{(a^{0.5} - b^{0.5})}} = \frac{a^{0.5} + b^{0.5}}{b^{0.5}}$.
Полученное выражение соответствует варианту ответа B).
Ответ: B) $\frac{a^{0.5} + b^{0.5}}{b^{0.5}}$
Другие задания:
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