gdz.top
Номер 751, страница 268 - гдз по алгебре 8 класс учебник Колягин, Ткачева
Авторы: Колягин Ю. М., Ткачева М. В., Федорова Н. Е., Шабунин М. И.
Тип: Учебник
Издательство: Просвещение
Год издания: 2023 - 2026
Уровень обучения: базовый
Цвет обложки: розовый, голубой
Популярные ГДЗ в 8 классе
Упражнения для повторения курса алгебры 8 класса - номер 751, страница 268.
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" => 207515 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "268" "field_page_end" => null "field_url" => "/8-klass/algebra/kolyagin-fgos/751" "field_display_title" => "751" "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" => 206300 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Упражнения для повторения курса алгебры 8 класса" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "256" "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 {#1082 #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" => 1399 "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" => 5457 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "8" "field_cases" => array:6 [ "field_accusative_case" => "восьмой" "field_creative_case" => "восьмым" "field_dative_case" => "восьмому" "field_genitive_case" => "восьмого" "field_nominative_case" => "восьмой" "field_prepositional_case" => "восьмом" ] "field_translit" => "vosmoj" ] #original: array:6 [ "id" => 5457 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "8" "field_cases" => array:6 [ "field_accusative_case" => "восьмой" "field_creative_case" => "восьмым" "field_dative_case" => "восьмому" "field_genitive_case" => "восьмого" "field_nominative_case" => "восьмой" "field_prepositional_case" => "восьмом" ] "field_translit" => "vosmoj" ] #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" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #original: array:6 [ "id" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #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:4 [ 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" => 4640 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Колягин" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Юрий" "field_patronymic" => "Михайлович" "field_surname_rp" => null ] #original: array:12 [ "id" => 4640 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Колягин" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Юрий" "field_patronymic" => "Михайлович" "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" => 5935 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ткачева" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Мария" "field_patronymic" => "Владимировна" "field_surname_rp" => null ] #original: array:12 [ "id" => 5935 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ткачева" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Мария" "field_patronymic" => "Владимировна" "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 => "*" ] } 2 => App\Models\Term {#1058 #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" => 6024 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Федорова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Надежда" "field_patronymic" => "Евгеньевна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6024 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Федорова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Надежда" "field_patronymic" => "Евгеньевна" "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 => "*" ] } 3 => 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:12 [ "id" => 6266 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Шабунин" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Михаил" "field_patronymic" => "Иванович" "field_surname_rp" => null ] #original: array:12 [ "id" => 6266 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Шабунин" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Михаил" "field_patronymic" => "Иванович" "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 {#1060 #items: [] #escapeWhenCastingToString: false } "field_book_type" => 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: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 {#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" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "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" => "rossiya" ] #original: array:6 [ "id" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "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" => "rossiya" ] #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 {#1065 #items: array:1 [ 0 => App\Models\Term {#1066 #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" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "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" => "moskva" ] #original: array:6 [ "id" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "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" => "moskva" ] #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 {#1067 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1069 #items: array:1 [ 0 => App\Models\Term {#1070 #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" => 38 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "базовый" "field_cases" => array:6 [ "field_accusative_case" => "базовый" "field_creative_case" => "базовым" "field_dative_case" => "базовому" "field_genitive_case" => "базового" "field_nominative_case" => "базовый" "field_prepositional_case" => "базовом" ] "field_translit" => "bazovyy" ] #original: array:6 [ "id" => 38 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "базовый" "field_cases" => array:6 [ "field_accusative_case" => "базовый" "field_creative_case" => "базовым" "field_dative_case" => "базовому" "field_genitive_case" => "базового" "field_nominative_case" => "базовый" "field_prepositional_case" => "базовом" ] "field_translit" => "bazovyy" ] #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_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1072 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1073 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2023" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное пособие" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1290" "field_priority" => "10" "field_default_folder" => "/algebra_08/kolyagin-u23/" "field_isbn" => null "field_cover" => array:1 [ 0 => "/media/algebra_08/kolyagin-u23/covers/cover1.webp?ts=1734206325" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_08/kolyagin-u23/covers/cover1.webp?ts=1734206325" "alt" => "" "width" => "1200" "height" => "1576" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1074 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1075 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1076 #items: [] #escapeWhenCastingToString: false } "field_url" => "/8-klass/algebra/kolyagin-fgos" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1077 #items: array:2 [ 0 => App\Models\Term {#1078 #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" => 53 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "розовый" "field_cases" => array:6 [ "field_accusative_case" => "розовый" "field_creative_case" => "розовым" "field_dative_case" => "розовому" "field_genitive_case" => "розового" "field_nominative_case" => "розовый" 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На работу с репетиром решаются не все, да и не у каждого родителя на это есть средства. В таких случаях на помощь приходит <strong>ГДЗ по алгебре 8 класс к учебнику Колягина (Просвещение, 2023 г.)</strong> с задачами и решениями. </p> <h2> Кому помогает сборник ответов? </h2> <p> Решебник полностью повторяет все темы, задачи и примеры, которые представлены в учебнике. Это позволяет намного быстрее и проще найти интересующие восьмиклассника решения. Книга будет полезна для: </p> <ul> <li>учеников, которые по причине олимпиад, соревнований или заболеваний, отсутствуют на уроках;</li> <li>подростков, за время урока не смогли освоить новый материал, поэтому затрудняются с выполнением домашнего задания;</li> <li>выпускников, которые готовятся к сдаче ЕГЭ или ОГЭ и им нужно заполнить пробелы в своих знаниях;</li> <li>учеников, которые не уверены в правильности ответа, и хотят проверить, правильно ли они выполнили упражнение;</li> <li>родителей, желающих проверить то, насколько их ребенок верно решил все задачи и примеры в домашнем задании.</li> </ul> <p> Как ни странно, решебниками, порой, пользуются и учителя при проверке классных или домашних работ, когда нет времени на проверку расчетов или последовательности действий в том или ином задании. Есть случаи, когда педагоги используют их, как пособие, при подготовке к уроку. </p> <h2> Польза готовых решений по учебнику Колягина </h2> <p> Использовать помощник при решении домашних заданий нужно правильно, не злоупотребляя информацией в нем. Работа с пособием должна быть направлена на помощь в учебе, а не систематическое списывание готовых решений с него. Такая методика может привести к тому, что ученик не сможет разобраться в пропущенной теме или заполнить пробелы в знаниях, а это в дальнейшем приведет к еще большим проблемам. В старших классах все темы в алгебре связаны между собой и если будет упущена одна, начнется недопонимание и в других. </p> <p> <strong>«ГДЗ по алгебре 8 класс Учебник Ю.М. Колягин, М.В. Ткачева, С.А. Федорова, М.И. Шабунин (ФГОС)»</strong> представляет вспомогательные и учебные материалы, которые предусмотрены на улучшение знаний ученика в школьной программе. Все больше педагогов и родителей соглашаются с тем, что пособие помогает: </p> <ol> <li>грамотно оформлять решения различных примеров и задач;</li> <li>получить навыки самопроверки и самоконтроля;</li> <li>выполнять домашнюю работу быстрее в условиях, когда время на выполнение решений ограничено;</li> <li>найти решение для сложных задач, при решении которых возникают трудности с поиском правильного решения и его грамотным оформлением;</li> <li>сверить полученный ответ с решебником, чтобы вовремя выявить ошибки, разобраться в них и оформить верный ответ.</li> </ol> <p> Помимо готового решения, к каждому готовому решению дается подробное объяснение, что особенно актуально для тех, кто не разбирается в материале или вообще пропустил урок. Дополнительная информация позволяет ученику углубиться в суть задачи и понять смысл каждого арифметического действия. Такой решебник очень полезен, если школьник неправильно решил задачу и хочет разобраться в допущенной ошибке, чтобы исключить ее повторение в будущем. Для ряда заданий есть несколько вариантов решений с пояснениями к каждому отдельно. Обычно это примеры или задачи, которые требуют нестандартного решения, которые отмечены «звездочкой». Иными словами, все упражнения расписаны максимально подробно, поэтому они будут понятны даже ученику, который по каким-либо отсутствовал на уроке. </p> <p> Пособие полностью соответствует новому стандарту ФГОС, по которому написан учебник. В ряде случаев оно помогает сэкономить время и силы школьника на поиск нужной информации. Использование готовых решений при выполнении домашних работ поможет повысить успеваемость и подтянуть свои знания по предмету. При этом ученику перед использованием ГДЗ рекомендуется повторить, теорию и правила, чтобы понимать о чем идет речь в решениях. </p> <h2> Почему ГДЗ по алгебре 8 класс так полезен? </h2> <p> Это пособие, которому можно доверять. Изначально оно успело зарекомендовать себя в печатном варианте, так как учебник используется в программе восьмиклассников уже давно. Онлайн версия помогает намного быстрее найти ответ с подробным решением намного быстрее. При этом она доступна в любое время, даже если школьник находится далеко от дома. Он может в любое время обратиться за помощью к электронной версии, так как современный гаджет с доступом в интернет есть у каждого ученика. </p> <p> ГДЗ составлен опытными методистами, которые имеют многолетний опыт в преподавании алгебры. Перед публикацией пособие проверено на соответствие учебнику по ФГОС, чтобы исключить некачественный контент. Решебник помог уже нескольким тысячам учеников, педагогов и родителей, что подтверждено многочисленными положительными отзывами. </p>" "breadcrumbs" => array:3 [ "class" => 379 "subject" => 542 "class_subject" => 385 ] ] #original: array:50 [ "id" => 1399 "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 {#1060} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1061} "field_country" => Illuminate\Database\Eloquent\Collection {#1063} "field_city" => Illuminate\Database\Eloquent\Collection {#1065} "field_series" => Illuminate\Database\Eloquent\Collection {#1067} "field_umk" => Illuminate\Database\Eloquent\Collection {#1068} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1069} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1071} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1072} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1073} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2023" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное пособие" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1290" "field_priority" => "10" "field_default_folder" => "/algebra_08/kolyagin-u23/" "field_isbn" => null "field_cover" => array:1 [ 0 => "/media/algebra_08/kolyagin-u23/covers/cover1.webp?ts=1734206325" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_08/kolyagin-u23/covers/cover1.webp?ts=1734206325" "alt" => "" "width" => "1200" "height" => "1576" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1074} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1075} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1076} "field_url" => "/8-klass/algebra/kolyagin-fgos" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1077} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => "<!--Текст в низу страницы--> <p> Есть случаи, когда по каким-либо причинам, ученик не может справиться с домашним заданием по алгебре или недостаточно освоил новую тему. На работу с репетиром решаются не все, да и не у каждого родителя на это есть средства. В таких случаях на помощь приходит <strong>ГДЗ по алгебре 8 класс к учебнику Колягина (Просвещение, 2023 г.)</strong> с задачами и решениями. </p> <h2> Кому помогает сборник ответов? </h2> <p> Решебник полностью повторяет все темы, задачи и примеры, которые представлены в учебнике. Это позволяет намного быстрее и проще найти интересующие восьмиклассника решения. Книга будет полезна для: </p> <ul> <li>учеников, которые по причине олимпиад, соревнований или заболеваний, отсутствуют на уроках;</li> <li>подростков, за время урока не смогли освоить новый материал, поэтому затрудняются с выполнением домашнего задания;</li> <li>выпускников, которые готовятся к сдаче ЕГЭ или ОГЭ и им нужно заполнить пробелы в своих знаниях;</li> <li>учеников, которые не уверены в правильности ответа, и хотят проверить, правильно ли они выполнили упражнение;</li> <li>родителей, желающих проверить то, насколько их ребенок верно решил все задачи и примеры в домашнем задании.</li> </ul> <p> Как ни странно, решебниками, порой, пользуются и учителя при проверке классных или домашних работ, когда нет времени на проверку расчетов или последовательности действий в том или ином задании. Есть случаи, когда педагоги используют их, как пособие, при подготовке к уроку. </p> <h2> Польза готовых решений по учебнику Колягина </h2> <p> Использовать помощник при решении домашних заданий нужно правильно, не злоупотребляя информацией в нем. Работа с пособием должна быть направлена на помощь в учебе, а не систематическое списывание готовых решений с него. Такая методика может привести к тому, что ученик не сможет разобраться в пропущенной теме или заполнить пробелы в знаниях, а это в дальнейшем приведет к еще большим проблемам. В старших классах все темы в алгебре связаны между собой и если будет упущена одна, начнется недопонимание и в других. </p> <p> <strong>«ГДЗ по алгебре 8 класс Учебник Ю.М. Колягин, М.В. Ткачева, С.А. Федорова, М.И. Шабунин (ФГОС)»</strong> представляет вспомогательные и учебные материалы, которые предусмотрены на улучшение знаний ученика в школьной программе. Все больше педагогов и родителей соглашаются с тем, что пособие помогает: </p> <ol> <li>грамотно оформлять решения различных примеров и задач;</li> <li>получить навыки самопроверки и самоконтроля;</li> <li>выполнять домашнюю работу быстрее в условиях, когда время на выполнение решений ограничено;</li> <li>найти решение для сложных задач, при решении которых возникают трудности с поиском правильного решения и его грамотным оформлением;</li> <li>сверить полученный ответ с решебником, чтобы вовремя выявить ошибки, разобраться в них и оформить верный ответ.</li> </ol> <p> Помимо готового решения, к каждому готовому решению дается подробное объяснение, что особенно актуально для тех, кто не разбирается в материале или вообще пропустил урок. Дополнительная информация позволяет ученику углубиться в суть задачи и понять смысл каждого арифметического действия. Такой решебник очень полезен, если школьник неправильно решил задачу и хочет разобраться в допущенной ошибке, чтобы исключить ее повторение в будущем. Для ряда заданий есть несколько вариантов решений с пояснениями к каждому отдельно. Обычно это примеры или задачи, которые требуют нестандартного решения, которые отмечены «звездочкой». Иными словами, все упражнения расписаны максимально подробно, поэтому они будут понятны даже ученику, который по каким-либо отсутствовал на уроке. </p> <p> Пособие полностью соответствует новому стандарту ФГОС, по которому написан учебник. В ряде случаев оно помогает сэкономить время и силы школьника на поиск нужной информации. Использование готовых решений при выполнении домашних работ поможет повысить успеваемость и подтянуть свои знания по предмету. При этом ученику перед использованием ГДЗ рекомендуется повторить, теорию и правила, чтобы понимать о чем идет речь в решениях. </p> <h2> Почему ГДЗ по алгебре 8 класс так полезен? </h2> <p> Это пособие, которому можно доверять. Изначально оно успело зарекомендовать себя в печатном варианте, так как учебник используется в программе восьмиклассников уже давно. Онлайн версия помогает намного быстрее найти ответ с подробным решением намного быстрее. При этом она доступна в любое время, даже если школьник находится далеко от дома. Он может в любое время обратиться за помощью к электронной версии, так как современный гаджет с доступом в интернет есть у каждого ученика. </p> <p> ГДЗ составлен опытными методистами, которые имеют многолетний опыт в преподавании алгебры. Перед публикацией пособие проверено на соответствие учебнику по ФГОС, чтобы исключить некачественный контент. Решебник помог уже нескольким тысячам учеников, педагогов и родителей, что подтверждено многочисленными положительными отзывами. </p>" "breadcrumbs" => array:3 [ "class" => 379 "subject" => 542 "class_subject" => 385 ] ] #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 {#1080 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 206300 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Упражнения для повторения курса алгебры 8 класса" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047} "field_page_start" => "256" "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 {#1082} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1080} ] #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 {#1087 #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" => 230787 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1096 #items: array:1 [ 0 => App\Models\Edition {#1088 #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" => 715 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1090 #items: array:1 [ 0 => App\Models\Term {#1089 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] …24 } ] #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" => "tolik" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1091 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1092 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1093 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 715 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1090} "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" => "tolik" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1091} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1092} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1093} "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" => "207515" "type" => "task" ] "text" => "<p>Выполнить действия (751–753).</p><p><strong>751. 1)</strong> $(m^2 + \frac{1}{m^2} + 2) : (m + \frac{1}{m}) - \frac{m^3}{m^2 - 1};$</p><p><strong>2)</strong> $\frac{x^2 + y^2}{x} : (x^3 + \frac{y^4}{x} + 2xy^2) - \frac{1}{x^2y^2}.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "751-1.jpg" "alt" => null "width" => "1068" "height" => 399 "path" => "/media/algebra_08/kolyagin-u23/0-00/751-1.webp?ts=1734247553" ] ] ] #original: array:7 [ "id" => 230787 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1096} "task" => array:2 [ "refs" => "207515" "type" => "task" ] "text" => "<p>Выполнить действия (751–753).</p><p><strong>751. 1)</strong> $(m^2 + \frac{1}{m^2} + 2) : (m + \frac{1}{m}) - \frac{m^3}{m^2 - 1};$</p><p><strong>2)</strong> $\frac{x^2 + y^2}{x} : (x^3 + \frac{y^4}{x} + 2xy^2) - \frac{1}{x^2y^2}.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "751-1.jpg" "alt" => null "width" => "1068" "height" => 399 "path" => "/media/algebra_08/kolyagin-u23/0-00/751-1.webp?ts=1734247553" ] ] ] #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 {#1094 #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" => 230995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1104 #items: array:1 [ 0 => App\Models\Edition {#1095 #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" => 716 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 1" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1098 #items: array:1 [ 0 => App\Models\Term {#1097 …30} ] #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" => "Пелеванюк" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1099 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1100 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1101 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 716 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 1" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1098} "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" => "Пелеванюк" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1099} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1100} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1101} "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" => "207515" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "751-1.jpg" "alt" => null "width" => "1668" "height" => 1738 "path" => "/media/algebra_08/kolyagin-u23/1-00/751-1.webp?ts=1734247033" ] ] ] #original: array:6 [ "id" => 230995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1104} "task" => array:2 [ "refs" => "207515" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "751-1.jpg" "alt" => null "width" => "1668" "height" => 1738 "path" => "/media/algebra_08/kolyagin-u23/1-00/751-1.webp?ts=1734247033" ] ] ] #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 {#1102 #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" => 1417434 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1112 #items: array:1 [ 0 => App\Models\Edition {#1103 #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" => 5309 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 4" "field_order" => "5" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1106 #items: array:1 [ 0 => App\Models\Term {#1105 …30} ] #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" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "4-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1107 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1108 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1109 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5309 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 4" "field_order" => "5" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1106} "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" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "4-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1107} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1108} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1109} "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" => "207515" "type" => "task" ] "text" => "<p><strong>1)</strong> $\left( m^2 + \frac{1}{m^2} + 2 \right) : \left( m + \frac{1}{m} \right) - \frac{m^3}{m^2 - 1}$</p><p>Решим по действиям. Сначала упростим выражение в первой скобке. Заметим, что это полный квадрат суммы, так как $a^2+2ab+b^2=(a+b)^2$.</p><p>$m^2 + \frac{1}{m^2} + 2 = m^2 + 2 \cdot m \cdot \frac{1}{m} + \left(\frac{1}{m}\right)^2 = \left(m + \frac{1}{m}\right)^2$</p><p>Теперь выполним деление:</p><p>$\left(m^2 + \frac{1}{m^2} + 2\right) : \left(m + \frac{1}{m}\right) = \left(m + \frac{1}{m}\right)^2 : \left(m + \frac{1}{m}\right) = m + \frac{1}{m}$</p><p>После выполнения первого действия выражение принимает вид:</p><p>$\left(m + \frac{1}{m}\right) - \frac{m^3}{m^2 - 1}$</p><p>Приведем выражение к общему знаменателю. Сначала преобразуем первое слагаемое:</p><p>$m + \frac{1}{m} = \frac{m \cdot m}{m} + \frac{1}{m} = \frac{m^2 + 1}{m}$</p><p>Теперь выполним вычитание дробей, приведя их к общему знаменателю $m(m^2 - 1)$:</p><p>$\frac{m^2 + 1}{m} - \frac{m^3}{m^2 - 1} = \frac{(m^2 + 1)(m^2 - 1)}{m(m^2 - 1)} - \frac{m^3 \cdot m}{m(m^2 - 1)}$</p><p>В числителе первой дроби используем формулу разности квадратов $(a+b)(a-b)=a^2-b^2$, а во второй дроби перемножим степени:</p><p>$\frac{(m^2)^2 - 1^2}{m(m^2 - 1)} - \frac{m^4}{m(m^2 - 1)} = \frac{m^4 - 1}{m(m^2 - 1)} - \frac{m^4}{m(m^2 - 1)} = \frac{m^4 - 1 - m^4}{m(m^2 - 1)} = \frac{-1}{m(m^2 - 1)}$</p><p>Ответ: $-\frac{1}{m(m^2-1)}$</p><p><strong>2)</strong> $\frac{x^2 + y^2}{x} : \left( x^3 + \frac{y^4}{x} + 2xy^2 \right) - \frac{1}{x^2y^2}$</p><p>Решим по действиям. Сначала упростим выражение в скобках (делитель), приведя все слагаемые к общему знаменателю $x$:</p><p>$x^3 + \frac{y^4}{x} + 2xy^2 = \frac{x^3 \cdot x}{x} + \frac{y^4}{x} + \frac{2xy^2 \cdot x}{x} = \frac{x^4 + y^4 + 2x^2y^2}{x}$</p><p>Числитель $x^4 + 2x^2y^2 + y^4$ является полным квадратом суммы $(x^2+y^2)^2$:</p><p>$\frac{x^4 + 2x^2y^2 + y^4}{x} = \frac{(x^2 + y^2)^2}{x}$</p><p>Теперь выполним деление. Чтобы разделить на дробь, нужно умножить на обратную ей дробь:</p><p>$\frac{x^2 + y^2}{x} : \frac{(x^2 + y^2)^2}{x} = \frac{x^2 + y^2}{x} \cdot \frac{x}{(x^2 + y^2)^2}$</p><p>Сокращаем общие множители $x$ и $(x^2+y^2)$:</p><p>$\frac{1}{1} \cdot \frac{1}{x^2 + y^2} = \frac{1}{x^2 + y^2}$</p><p>Теперь выполним вычитание:</p><p>$\frac{1}{x^2 + y^2} - \frac{1}{x^2y^2}$</p><p>Приведем дроби к общему знаменателю $x^2y^2(x^2 + y^2)$:</p><p>$\frac{1 \cdot x^2y^2}{x^2y^2(x^2 + y^2)} - \frac{1 \cdot (x^2 + y^2)}{x^2y^2(x^2 + y^2)} = \frac{x^2y^2 - (x^2 + y^2)}{x^2y^2(x^2 + y^2)} = \frac{x^2y^2 - x^2 - y^2}{x^2y^2(x^2 + y^2)}$</p><p>Ответ: $\frac{x^2y^2 - x^2 - y^2}{x^2y^2(x^2 + y^2)}$</p>" ] #original: array:6 [ "id" => 1417434 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1112} "task" => array:2 [ "refs" => "207515" "type" => "task" ] "text" => "<p><strong>1)</strong> $\left( m^2 + \frac{1}{m^2} + 2 \right) : \left( m + \frac{1}{m} \right) - \frac{m^3}{m^2 - 1}$</p><p>Решим по действиям. Сначала упростим выражение в первой скобке. Заметим, что это полный квадрат суммы, так как $a^2+2ab+b^2=(a+b)^2$.</p><p>$m^2 + \frac{1}{m^2} + 2 = m^2 + 2 \cdot m \cdot \frac{1}{m} + \left(\frac{1}{m}\right)^2 = \left(m + \frac{1}{m}\right)^2$</p><p>Теперь выполним деление:</p><p>$\left(m^2 + \frac{1}{m^2} + 2\right) : \left(m + \frac{1}{m}\right) = \left(m + \frac{1}{m}\right)^2 : \left(m + \frac{1}{m}\right) = m + \frac{1}{m}$</p><p>После выполнения первого действия выражение принимает вид:</p><p>$\left(m + \frac{1}{m}\right) - \frac{m^3}{m^2 - 1}$</p><p>Приведем выражение к общему знаменателю. Сначала преобразуем первое слагаемое:</p><p>$m + \frac{1}{m} = \frac{m \cdot m}{m} + \frac{1}{m} = \frac{m^2 + 1}{m}$</p><p>Теперь выполним вычитание дробей, приведя их к общему знаменателю $m(m^2 - 1)$:</p><p>$\frac{m^2 + 1}{m} - \frac{m^3}{m^2 - 1} = \frac{(m^2 + 1)(m^2 - 1)}{m(m^2 - 1)} - \frac{m^3 \cdot m}{m(m^2 - 1)}$</p><p>В числителе первой дроби используем формулу разности квадратов $(a+b)(a-b)=a^2-b^2$, а во второй дроби перемножим степени:</p><p>$\frac{(m^2)^2 - 1^2}{m(m^2 - 1)} - \frac{m^4}{m(m^2 - 1)} = \frac{m^4 - 1}{m(m^2 - 1)} - \frac{m^4}{m(m^2 - 1)} = \frac{m^4 - 1 - m^4}{m(m^2 - 1)} = \frac{-1}{m(m^2 - 1)}$</p><p>Ответ: $-\frac{1}{m(m^2-1)}$</p><p><strong>2)</strong> $\frac{x^2 + y^2}{x} : \left( x^3 + \frac{y^4}{x} + 2xy^2 \right) - \frac{1}{x^2y^2}$</p><p>Решим по действиям. Сначала упростим выражение в скобках (делитель), приведя все слагаемые к общему знаменателю $x$:</p><p>$x^3 + \frac{y^4}{x} + 2xy^2 = \frac{x^3 \cdot x}{x} + \frac{y^4}{x} + \frac{2xy^2 \cdot x}{x} = \frac{x^4 + y^4 + 2x^2y^2}{x}$</p><p>Числитель $x^4 + 2x^2y^2 + y^4$ является полным квадратом суммы $(x^2+y^2)^2$:</p><p>$\frac{x^4 + 2x^2y^2 + y^4}{x} = \frac{(x^2 + y^2)^2}{x}$</p><p>Теперь выполним деление. Чтобы разделить на дробь, нужно умножить на обратную ей дробь:</p><p>$\frac{x^2 + y^2}{x} : \frac{(x^2 + y^2)^2}{x} = \frac{x^2 + y^2}{x} \cdot \frac{x}{(x^2 + y^2)^2}$</p><p>Сокращаем общие множители $x$ и $(x^2+y^2)$:</p><p>$\frac{1}{1} \cdot \frac{1}{x^2 + y^2} = \frac{1}{x^2 + y^2}$</p><p>Теперь выполним вычитание:</p><p>$\frac{1}{x^2 + y^2} - \frac{1}{x^2y^2}$</p><p>Приведем дроби к общему знаменателю $x^2y^2(x^2 + y^2)$:</p><p>$\frac{1 \cdot x^2y^2}{x^2y^2(x^2 + y^2)} - \frac{1 \cdot (x^2 + y^2)}{x^2y^2(x^2 + y^2)} = \frac{x^2y^2 - (x^2 + y^2)}{x^2y^2(x^2 + y^2)} = \frac{x^2y^2 - x^2 - y^2}{x^2y^2(x^2 + y^2)}$</p><p>Ответ: $\frac{x^2y^2 - x^2 - y^2}{x^2y^2(x^2 + y^2)}$</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" => "207516" "type" => "task" ] "previous" => array:2 [ "refs" => "207514" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1084 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1081 #items: array:1 [ 0 => App\Models\BookPage {#1040 #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" => 929734 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "268" "field_url" => "/8-klass/algebra/kolyagin-fgos/page-268" "field_display_title" => "268" "field_folder" => "1" "field_image_name" => "268" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1043 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1110 #items: array:1 [ 0 => App\Models\Book {#1048} ] #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 {#1111 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1122 #items: array:1 [ 0 => App\Models\Element {#1121 #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" => 1104126 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1123 #items: array:1 [ 0 => App\Models\Edition {#1088} ] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "929734" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "268-1.jpg" "alt" => null "width" => "1761" "height" => 2502 "path" => "/media/algebra_08/kolyagin-u23/0-1/268-1.webp?ts=1744701200" ] ] ] #original: array:6 [ "id" => 1104126 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1123} "book_page" => array:2 [ "refs" => "929734" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "268-1.jpg" "alt" => null "width" => "1761" "height" => 2502 "path" => "/media/algebra_08/kolyagin-u23/0-1/268-1.webp?ts=1744701200" ] ] ] #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" => "929735" "type" => "book_page" ] "previous" => array:2 [ "refs" => "929733" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1186 #items: array:7 [ 0 => App\Models\Task {#1200 #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" => 207509 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "268" "field_page_end" => null "field_url" => "/8-klass/algebra/kolyagin-fgos/745" "field_display_title" => "745" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1201 #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 {#1202 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1203 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1204 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1207 #items: array:3 [ 0 => App\Models\Element {#1216 …30} 1 => App\Models\Element {#1218 …30} 2 => App\Models\Element {#1225 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "207510" "type" => "task" ] "previous" => array:2 [ "refs" => "207508" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1213 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "929734" "type" => "book_page" ] ] #original: array:24 [ "id" => 207509 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "268" "field_page_end" => null "field_url" => "/8-klass/algebra/kolyagin-fgos/745" "field_display_title" => "745" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1201} "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 {#1202} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1203} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1204} "content" => Illuminate\Database\Eloquent\Collection {#1207} "next" => array:2 [ "refs" => "207510" "type" => "task" ] "previous" => array:2 [ "refs" => "207508" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1213} "page" => array:2 [ "refs" => "929734" "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 {#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" => 207510 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "268" "field_page_end" => null "field_url" => "/8-klass/algebra/kolyagin-fgos/746" "field_display_title" => "746" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1205 #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 {#1212 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1214 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1210 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1209 #items: array:3 [ 0 => App\Models\Element {#1236 …30} 1 => App\Models\Element {#1238 …30} 2 => App\Models\Element {#1240 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "207511" "type" => "task" ] "previous" => array:2 [ "refs" => "207509" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1233 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "929734" "type" => "book_page" ] ] #original: array:24 [ "id" => 207510 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "268" "field_page_end" => null "field_url" => "/8-klass/algebra/kolyagin-fgos/746" "field_display_title" => "746" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1205} "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 {#1212} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1214} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1210} "content" => Illuminate\Database\Eloquent\Collection {#1209} "next" => array:2 [ "refs" => "207511" "type" => "task" ] "previous" => array:2 [ "refs" => "207509" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1233} "page" => array:2 [ "refs" => "929734" "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 {#1211 #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" => 207511 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "268" "field_page_end" => null "field_url" => "/8-klass/algebra/kolyagin-fgos/747" "field_display_title" => "747" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1208 #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 {#1232 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1234 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1230 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1229 #items: array:3 [ 0 => App\Models\Element {#1250 …30} 1 => App\Models\Element {#1252 …30} 2 => App\Models\Element {#1254 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "207512" "type" => "task" ] "previous" => array:2 [ "refs" => "207510" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1247 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "929734" "type" => "book_page" ] ] #original: array:24 [ "id" => 207511 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "268" "field_page_end" => null "field_url" => "/8-klass/algebra/kolyagin-fgos/747" "field_display_title" => "747" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1208} "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 {#1232} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1234} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1230} "content" => Illuminate\Database\Eloquent\Collection {#1229} "next" => array:2 [ "refs" => "207512" "type" => "task" ] "previous" => array:2 [ "refs" => "207510" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1247} "page" => array:2 [ "refs" => "929734" "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 {#1231 #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" => 207512 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "268" "field_page_end" => null "field_url" => "/8-klass/algebra/kolyagin-fgos/748" "field_display_title" => "748" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1228 #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 {#1246 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1248 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1244 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1243 #items: array:3 [ 0 => App\Models\Element {#1264 …30} 1 => App\Models\Element {#1266 …30} 2 => App\Models\Element {#1268 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "207513" "type" => "task" ] "previous" => array:2 [ "refs" => "207511" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1261 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "929734" "type" => "book_page" ] ] #original: array:24 [ "id" => 207512 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "268" "field_page_end" => null "field_url" => "/8-klass/algebra/kolyagin-fgos/748" "field_display_title" => "748" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1228} "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 {#1246} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1248} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1244} "content" => Illuminate\Database\Eloquent\Collection {#1243} "next" => array:2 [ "refs" => "207513" "type" => "task" ] "previous" => array:2 [ "refs" => "207511" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1261} "page" => array:2 [ "refs" => "929734" "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" => 207513 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "268" "field_page_end" => null "field_url" => "/8-klass/algebra/kolyagin-fgos/749" "field_display_title" => "749" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1242 #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 {#1260 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1262 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1258 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1257 #items: array:3 [ 0 => App\Models\Element {#1278 …30} 1 => App\Models\Element {#1280 …30} 2 => App\Models\Element {#1282 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "207514" "type" => "task" ] "previous" => array:2 [ "refs" => "207512" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1275 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "929734" "type" => "book_page" ] ] #original: array:24 [ "id" => 207513 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "268" "field_page_end" => null "field_url" => "/8-klass/algebra/kolyagin-fgos/749" "field_display_title" => "749" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1242} "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 {#1260} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1262} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1258} "content" => Illuminate\Database\Eloquent\Collection {#1257} "next" => array:2 [ "refs" => "207514" "type" => "task" ] "previous" => array:2 [ "refs" => "207512" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1275} "page" => array:2 [ "refs" => "929734" "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 {#1259 #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" => 207514 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "268" "field_page_end" => null "field_url" => "/8-klass/algebra/kolyagin-fgos/750" "field_display_title" => "750" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1256 #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 {#1274 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1276 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1272 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1273 #items: array:2 [ 0 => App\Models\Element {#1291 …30} 1 => App\Models\Element {#1293 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "207515" "type" => "task" ] "previous" => array:2 [ "refs" => "207513" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1288 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => 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#observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№751 (с. 268)
Условие. №751 (с. 268)
скриншот условия
Выполнить действия (751–753).
751. 1) $(m^2 + \frac{1}{m^2} + 2) : (m + \frac{1}{m}) - \frac{m^3}{m^2 - 1};$
2) $\frac{x^2 + y^2}{x} : (x^3 + \frac{y^4}{x} + 2xy^2) - \frac{1}{x^2y^2}.$
Решение 1. №751 (с. 268)
Решение 4. №751 (с. 268)
1) $\left( m^2 + \frac{1}{m^2} + 2 \right) : \left( m + \frac{1}{m} \right) - \frac{m^3}{m^2 - 1}$
Решим по действиям. Сначала упростим выражение в первой скобке. Заметим, что это полный квадрат суммы, так как $a^2+2ab+b^2=(a+b)^2$.
$m^2 + \frac{1}{m^2} + 2 = m^2 + 2 \cdot m \cdot \frac{1}{m} + \left(\frac{1}{m}\right)^2 = \left(m + \frac{1}{m}\right)^2$
Теперь выполним деление:
$\left(m^2 + \frac{1}{m^2} + 2\right) : \left(m + \frac{1}{m}\right) = \left(m + \frac{1}{m}\right)^2 : \left(m + \frac{1}{m}\right) = m + \frac{1}{m}$
После выполнения первого действия выражение принимает вид:
$\left(m + \frac{1}{m}\right) - \frac{m^3}{m^2 - 1}$
Приведем выражение к общему знаменателю. Сначала преобразуем первое слагаемое:
$m + \frac{1}{m} = \frac{m \cdot m}{m} + \frac{1}{m} = \frac{m^2 + 1}{m}$
Теперь выполним вычитание дробей, приведя их к общему знаменателю $m(m^2 - 1)$:
$\frac{m^2 + 1}{m} - \frac{m^3}{m^2 - 1} = \frac{(m^2 + 1)(m^2 - 1)}{m(m^2 - 1)} - \frac{m^3 \cdot m}{m(m^2 - 1)}$
В числителе первой дроби используем формулу разности квадратов $(a+b)(a-b)=a^2-b^2$, а во второй дроби перемножим степени:
$\frac{(m^2)^2 - 1^2}{m(m^2 - 1)} - \frac{m^4}{m(m^2 - 1)} = \frac{m^4 - 1}{m(m^2 - 1)} - \frac{m^4}{m(m^2 - 1)} = \frac{m^4 - 1 - m^4}{m(m^2 - 1)} = \frac{-1}{m(m^2 - 1)}$
Ответ: $-\frac{1}{m(m^2-1)}$
2) $\frac{x^2 + y^2}{x} : \left( x^3 + \frac{y^4}{x} + 2xy^2 \right) - \frac{1}{x^2y^2}$
Решим по действиям. Сначала упростим выражение в скобках (делитель), приведя все слагаемые к общему знаменателю $x$:
$x^3 + \frac{y^4}{x} + 2xy^2 = \frac{x^3 \cdot x}{x} + \frac{y^4}{x} + \frac{2xy^2 \cdot x}{x} = \frac{x^4 + y^4 + 2x^2y^2}{x}$
Числитель $x^4 + 2x^2y^2 + y^4$ является полным квадратом суммы $(x^2+y^2)^2$:
$\frac{x^4 + 2x^2y^2 + y^4}{x} = \frac{(x^2 + y^2)^2}{x}$
Теперь выполним деление. Чтобы разделить на дробь, нужно умножить на обратную ей дробь:
$\frac{x^2 + y^2}{x} : \frac{(x^2 + y^2)^2}{x} = \frac{x^2 + y^2}{x} \cdot \frac{x}{(x^2 + y^2)^2}$
Сокращаем общие множители $x$ и $(x^2+y^2)$:
$\frac{1}{1} \cdot \frac{1}{x^2 + y^2} = \frac{1}{x^2 + y^2}$
Теперь выполним вычитание:
$\frac{1}{x^2 + y^2} - \frac{1}{x^2y^2}$
Приведем дроби к общему знаменателю $x^2y^2(x^2 + y^2)$:
$\frac{1 \cdot x^2y^2}{x^2y^2(x^2 + y^2)} - \frac{1 \cdot (x^2 + y^2)}{x^2y^2(x^2 + y^2)} = \frac{x^2y^2 - (x^2 + y^2)}{x^2y^2(x^2 + y^2)} = \frac{x^2y^2 - x^2 - y^2}{x^2y^2(x^2 + y^2)}$
Ответ: $\frac{x^2y^2 - x^2 - y^2}{x^2y^2(x^2 + y^2)}$
Другие задания:
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