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Номер 285, страница 87 - гдз по алгебре 9 класс учебник Солтан, Солтан
Авторы: Солтан Г. Н., Солтан А. Е., Жумадилова А. Ж.
Тип: Учебник
Издательство: Кокшетау
Год издания: 2019 - 2026
ISBN: 978-601-317-424-2
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 9 классе
II. Элементы комбинаторики. 11. Бином Ньютона и его свойства - номер 285, страница 87.
App\Models\Task {#1024 // 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" => 1108001 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "87" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-285" "field_display_title" => "285" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331 #items: array:1 [ 0 => App\Models\Term {#1029 #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 {#1026 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342 #items: array:1 [ 0 => App\Models\Branch {#1023 #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" => 1107646 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "II. Элементы комбинаторики" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "62" "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 {#1033 #items: array:1 [ 0 => App\Models\Book {#1030 #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" => 4298 "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 {#1032 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1035 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1039 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1050 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4298 "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 {#1032 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1035 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1039 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1050 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1060 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107646 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "II. 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Бином Ньютона и его свойства" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "82" "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 {#1064 #items: array:1 [ 0 => App\Models\Book {#1065 #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" => 4298 "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 {#1066 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1072 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1077 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1079 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1081 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1083 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4298 "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 {#1066 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1072 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1077 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1079 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1081 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1083 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1093 #items: array:1 [ 0 => App\Models\Branch {#1094 #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" => 1107646 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "II. Элементы комбинаторики" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_page_start" => "62" "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 {#1096 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1125 …2} ] #original: array:24 [ "id" => 1107646 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "II. Элементы комбинаторики" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_page_start" => "62" "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 {#1096 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1125 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107651 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "11. Бином Ньютона и его свойства" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063} "field_page_start" => "82" "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 {#1064} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1093} ] #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 {#1126 #items: array:3 [ 0 => App\Models\Element {#1127 #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" => 1276640 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1128 #items: array:1 [ 0 => App\Models\Edition {#1129 #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" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1130 …2} "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 {#1132 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1133 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1134 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1130 …2} "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 {#1132 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1133 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1134 …2} "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" => "1108001" "type" => "task" ] "text" => "<p><strong>285.</strong> Вычислите:</p><p><strong>а)</strong> $(\sqrt{5} + \sqrt{2})^5$;</p><p><strong>б)</strong> $(\sqrt{3} - 1)^6$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "285-1.jpg" "alt" => null "width" => "1059" "height" => 154 "path" => "/media/algebra_09/soltan-u19/0-1/285-1.webp?ts=1752834333" ] ] ] #original: array:7 [ "id" => 1276640 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1128} "task" => array:2 [ "refs" => "1108001" "type" => "task" ] "text" => "<p><strong>285.</strong> Вычислите:</p><p><strong>а)</strong> $(\sqrt{5} + \sqrt{2})^5$;</p><p><strong>б)</strong> $(\sqrt{3} - 1)^6$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "285-1.jpg" "alt" => null "width" => "1059" "height" => 154 "path" => "/media/algebra_09/soltan-u19/0-1/285-1.webp?ts=1752834333" ] ] ] #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 {#1135 #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" => 1277931 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1136 #items: array:1 [ 0 => App\Models\Edition {#1137 #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" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1138 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1140 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1141 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1142 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1138 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1140 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1141 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1142 …2} "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" => "1108001" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "285-1.jpg" "alt" => null "width" => "1301" "height" => 1059 "path" => "/media/algebra_09/soltan-u19/1-1/285-1.webp?ts=1752831032" ] ] ] #original: array:6 [ "id" => 1277931 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1136} "task" => array:2 [ "refs" => "1108001" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "285-1.jpg" "alt" => null "width" => "1301" "height" => 1059 "path" => "/media/algebra_09/soltan-u19/1-1/285-1.webp?ts=1752831032" ] ] ] #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 {#1143 #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" => 1553886 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1144 #items: array:1 [ 0 => App\Models\Edition {#1145 #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" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1146 …2} "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 {#1148 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1149 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1150 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1146 …2} "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 {#1148 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1149 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1150 …2} "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" => "1108001" "type" => "task" ] "text" => "<p><strong>а)</strong> Для вычисления $(\sqrt{5} + \sqrt{2})^5$ воспользуемся формулой бинома Ньютона:</p><p>$(a+b)^n = C_n^0 a^n + C_n^1 a^{n-1}b + C_n^2 a^{n-2}b^2 + \dots + C_n^n b^n$</p><p>В нашем случае $a = \sqrt{5}$, $b = \sqrt{2}$, $n=5$. Биномиальные коэффициенты для $n=5$ равны: $C_5^0=1$, $C_5^1=5$, $C_5^2=10$, $C_5^3=10$, $C_5^4=5$, $C_5^5=1$.</p><p>Разложим выражение по формуле:</p><p>$(\sqrt{5} + \sqrt{2})^5 = C_5^0(\sqrt{5})^5 + C_5^1(\sqrt{5})^4(\sqrt{2})^1 + C_5^2(\sqrt{5})^3(\sqrt{2})^2 + C_5^3(\sqrt{5})^2(\sqrt{2})^3 + C_5^4(\sqrt{5})^1(\sqrt{2})^4 + C_5^5(\sqrt{2})^5$</p><p>Теперь вычислим каждый член разложения:</p><p>$C_5^0(\sqrt{5})^5 = 1 \cdot (\sqrt{5})^5 = (\sqrt{5}^2)^2 \cdot \sqrt{5} = 5^2 \cdot \sqrt{5} = 25\sqrt{5}$</p><p>$C_5^1(\sqrt{5})^4(\sqrt{2})^1 = 5 \cdot (\sqrt{5})^4 \cdot \sqrt{2} = 5 \cdot 5^2 \cdot \sqrt{2} = 5 \cdot 25 \cdot \sqrt{2} = 125\sqrt{2}$</p><p>$C_5^2(\sqrt{5})^3(\sqrt{2})^2 = 10 \cdot (5\sqrt{5}) \cdot 2 = 100\sqrt{5}$</p><p>$C_5^3(\sqrt{5})^2(\sqrt{2})^3 = 10 \cdot 5 \cdot (2\sqrt{2}) = 100\sqrt{2}$</p><p>$C_5^4(\sqrt{5})^1(\sqrt{2})^4 = 5 \cdot \sqrt{5} \cdot 4 = 20\sqrt{5}$</p><p>$C_5^5(\sqrt{2})^5 = 1 \cdot (\sqrt{2})^5 = (\sqrt{2}^2)^2 \cdot \sqrt{2} = 2^2 \cdot \sqrt{2} = 4\sqrt{2}$</p><p>Сложим все полученные значения:</p><p>$(\sqrt{5} + \sqrt{2})^5 = 25\sqrt{5} + 125\sqrt{2} + 100\sqrt{5} + 100\sqrt{2} + 20\sqrt{5} + 4\sqrt{2}$</p><p>Сгруппируем слагаемые с $\sqrt{5}$ и с $\sqrt{2}$:</p><p>$(25\sqrt{5} + 100\sqrt{5} + 20\sqrt{5}) + (125\sqrt{2} + 100\sqrt{2} + 4\sqrt{2}) = (25+100+20)\sqrt{5} + (125+100+4)\sqrt{2} = 145\sqrt{5} + 229\sqrt{2}$</p><p><strong>Ответ:</strong> $145\sqrt{5} + 229\sqrt{2}$</p><p><strong>б)</strong> Для вычисления $(\sqrt{3} - 1)^6$ также воспользуемся формулой бинома Ньютона для разности:</p><p>$(a-b)^n = C_n^0 a^n - C_n^1 a^{n-1}b + C_n^2 a^{n-2}b^2 - \dots + (-1)^n C_n^n b^n$</p><p>В данном случае $a = \sqrt{3}$, $b = 1$, $n=6$. Биномиальные коэффициенты для $n=6$ равны: $C_6^0=1$, $C_6^1=6$, $C_6^2=15$, $C_6^3=20$, $C_6^4=15$, $C_6^5=6$, $C_6^6=1$.</p><p>Разложим выражение по формуле, учитывая чередование знаков:</p><p>$(\sqrt{3} - 1)^6 = C_6^0(\sqrt{3})^6 - C_6^1(\sqrt{3})^5 \cdot 1^1 + C_6^2(\sqrt{3})^4 \cdot 1^2 - C_6^3(\sqrt{3})^3 \cdot 1^3 + C_6^4(\sqrt{3})^2 \cdot 1^4 - C_6^5(\sqrt{3})^1 \cdot 1^5 + C_6^6(\sqrt{3})^0 \cdot 1^6$</p><p>Вычислим каждый член разложения:</p><p>$C_6^0(\sqrt{3})^6 = 1 \cdot (\sqrt{3})^6 = 3^3 = 27$</p><p>$-C_6^1(\sqrt{3})^5 = -6 \cdot (\sqrt{3})^5 = -6 \cdot (3^2 \cdot \sqrt{3}) = -6 \cdot 9\sqrt{3} = -54\sqrt{3}$</p><p>$C_6^2(\sqrt{3})^4 = 15 \cdot (\sqrt{3})^4 = 15 \cdot 3^2 = 15 \cdot 9 = 135$</p><p>$-C_6^3(\sqrt{3})^3 = -20 \cdot (\sqrt{3})^3 = -20 \cdot (3\sqrt{3}) = -60\sqrt{3}$</p><p>$C_6^4(\sqrt{3})^2 = 15 \cdot (\sqrt{3})^2 = 15 \cdot 3 = 45$</p><p>$-C_6^5(\sqrt{3})^1 = -6 \cdot \sqrt{3} = -6\sqrt{3}$</p><p>$C_6^6(\sqrt{3})^0 = 1 \cdot (\sqrt{3})^0 \cdot 1^6 = 1 \cdot 1 \cdot 1 = 1$</p><p>Сложим все полученные значения:</p><p>$(\sqrt{3} - 1)^6 = 27 - 54\sqrt{3} + 135 - 60\sqrt{3} + 45 - 6\sqrt{3} + 1$</p><p>Сгруппируем рациональные и иррациональные слагаемые:</p><p>$(27 + 135 + 45 + 1) + (-54\sqrt{3} - 60\sqrt{3} - 6\sqrt{3}) = 208 - (54+60+6)\sqrt{3} = 208 - 120\sqrt{3}$</p><p><strong>Ответ:</strong> $208 - 120\sqrt{3}$</p>" ] #original: array:6 [ "id" => 1553886 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1144} "task" => array:2 [ "refs" => "1108001" "type" => "task" ] "text" => "<p><strong>а)</strong> Для вычисления $(\sqrt{5} + \sqrt{2})^5$ воспользуемся формулой бинома Ньютона:</p><p>$(a+b)^n = C_n^0 a^n + C_n^1 a^{n-1}b + C_n^2 a^{n-2}b^2 + \dots + C_n^n b^n$</p><p>В нашем случае $a = \sqrt{5}$, $b = \sqrt{2}$, $n=5$. Биномиальные коэффициенты для $n=5$ равны: $C_5^0=1$, $C_5^1=5$, $C_5^2=10$, $C_5^3=10$, $C_5^4=5$, $C_5^5=1$.</p><p>Разложим выражение по формуле:</p><p>$(\sqrt{5} + \sqrt{2})^5 = C_5^0(\sqrt{5})^5 + C_5^1(\sqrt{5})^4(\sqrt{2})^1 + C_5^2(\sqrt{5})^3(\sqrt{2})^2 + C_5^3(\sqrt{5})^2(\sqrt{2})^3 + C_5^4(\sqrt{5})^1(\sqrt{2})^4 + C_5^5(\sqrt{2})^5$</p><p>Теперь вычислим каждый член разложения:</p><p>$C_5^0(\sqrt{5})^5 = 1 \cdot (\sqrt{5})^5 = (\sqrt{5}^2)^2 \cdot \sqrt{5} = 5^2 \cdot \sqrt{5} = 25\sqrt{5}$</p><p>$C_5^1(\sqrt{5})^4(\sqrt{2})^1 = 5 \cdot (\sqrt{5})^4 \cdot \sqrt{2} = 5 \cdot 5^2 \cdot \sqrt{2} = 5 \cdot 25 \cdot \sqrt{2} = 125\sqrt{2}$</p><p>$C_5^2(\sqrt{5})^3(\sqrt{2})^2 = 10 \cdot (5\sqrt{5}) \cdot 2 = 100\sqrt{5}$</p><p>$C_5^3(\sqrt{5})^2(\sqrt{2})^3 = 10 \cdot 5 \cdot (2\sqrt{2}) = 100\sqrt{2}$</p><p>$C_5^4(\sqrt{5})^1(\sqrt{2})^4 = 5 \cdot \sqrt{5} \cdot 4 = 20\sqrt{5}$</p><p>$C_5^5(\sqrt{2})^5 = 1 \cdot (\sqrt{2})^5 = (\sqrt{2}^2)^2 \cdot \sqrt{2} = 2^2 \cdot \sqrt{2} = 4\sqrt{2}$</p><p>Сложим все полученные значения:</p><p>$(\sqrt{5} + \sqrt{2})^5 = 25\sqrt{5} + 125\sqrt{2} + 100\sqrt{5} + 100\sqrt{2} + 20\sqrt{5} + 4\sqrt{2}$</p><p>Сгруппируем слагаемые с $\sqrt{5}$ и с $\sqrt{2}$:</p><p>$(25\sqrt{5} + 100\sqrt{5} + 20\sqrt{5}) + (125\sqrt{2} + 100\sqrt{2} + 4\sqrt{2}) = (25+100+20)\sqrt{5} + (125+100+4)\sqrt{2} = 145\sqrt{5} + 229\sqrt{2}$</p><p><strong>Ответ:</strong> $145\sqrt{5} + 229\sqrt{2}$</p><p><strong>б)</strong> Для вычисления $(\sqrt{3} - 1)^6$ также воспользуемся формулой бинома Ньютона для разности:</p><p>$(a-b)^n = C_n^0 a^n - C_n^1 a^{n-1}b + C_n^2 a^{n-2}b^2 - \dots + (-1)^n C_n^n b^n$</p><p>В данном случае $a = \sqrt{3}$, $b = 1$, $n=6$. Биномиальные коэффициенты для $n=6$ равны: $C_6^0=1$, $C_6^1=6$, $C_6^2=15$, $C_6^3=20$, $C_6^4=15$, $C_6^5=6$, $C_6^6=1$.</p><p>Разложим выражение по формуле, учитывая чередование знаков:</p><p>$(\sqrt{3} - 1)^6 = C_6^0(\sqrt{3})^6 - C_6^1(\sqrt{3})^5 \cdot 1^1 + C_6^2(\sqrt{3})^4 \cdot 1^2 - C_6^3(\sqrt{3})^3 \cdot 1^3 + C_6^4(\sqrt{3})^2 \cdot 1^4 - C_6^5(\sqrt{3})^1 \cdot 1^5 + C_6^6(\sqrt{3})^0 \cdot 1^6$</p><p>Вычислим каждый член разложения:</p><p>$C_6^0(\sqrt{3})^6 = 1 \cdot (\sqrt{3})^6 = 3^3 = 27$</p><p>$-C_6^1(\sqrt{3})^5 = -6 \cdot (\sqrt{3})^5 = -6 \cdot (3^2 \cdot \sqrt{3}) = -6 \cdot 9\sqrt{3} = -54\sqrt{3}$</p><p>$C_6^2(\sqrt{3})^4 = 15 \cdot (\sqrt{3})^4 = 15 \cdot 3^2 = 15 \cdot 9 = 135$</p><p>$-C_6^3(\sqrt{3})^3 = -20 \cdot (\sqrt{3})^3 = -20 \cdot (3\sqrt{3}) = -60\sqrt{3}$</p><p>$C_6^4(\sqrt{3})^2 = 15 \cdot (\sqrt{3})^2 = 15 \cdot 3 = 45$</p><p>$-C_6^5(\sqrt{3})^1 = -6 \cdot \sqrt{3} = -6\sqrt{3}$</p><p>$C_6^6(\sqrt{3})^0 = 1 \cdot (\sqrt{3})^0 \cdot 1^6 = 1 \cdot 1 \cdot 1 = 1$</p><p>Сложим все полученные значения:</p><p>$(\sqrt{3} - 1)^6 = 27 - 54\sqrt{3} + 135 - 60\sqrt{3} + 45 - 6\sqrt{3} + 1$</p><p>Сгруппируем рациональные и иррациональные слагаемые:</p><p>$(27 + 135 + 45 + 1) + (-54\sqrt{3} - 60\sqrt{3} - 6\sqrt{3}) = 208 - (54+60+6)\sqrt{3} = 208 - 120\sqrt{3}$</p><p><strong>Ответ:</strong> $208 - 120\sqrt{3}$</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" => "1108002" "type" => "task" ] "previous" => array:2 [ "refs" => "1108000" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1151 #items: array:1 [ 0 => App\Models\Book {#1152 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false 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Illuminate\Database\Eloquent\Collection {#1191 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1193 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1195 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1199 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1200 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1202 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1204 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1206 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1207 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1208 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1209 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1210 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1211 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1212 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1213 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1214 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1215 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4298 "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 {#1189 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1191 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1193 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1195 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1199 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1200 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1202 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1204 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1206 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1207 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1208 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1209 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1210 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1211 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1212 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1213 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1214 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1215 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 "edition_groups" => Illuminate\Database\Eloquent\Collection {#1216 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1217 #items: array:1 [ 0 => App\Models\Element {#1218 #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" => 1260980 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1219 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1260980 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1219 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #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" => "1097830" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097828" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1226 #items: array:9 [ 0 => App\Models\Task {#1227 #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" => 1108000 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "87" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-284" "field_display_title" => "284" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1228 …2} "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 {#1230 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1231 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1264 …2} "content" => Illuminate\Database\Eloquent\Collection {#1361 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1380 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108000 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "87" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-284" "field_display_title" => "284" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1228 …2} "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 {#1230 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1231 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1264 …2} "content" => Illuminate\Database\Eloquent\Collection {#1361 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1380 …2} "page" => array:2 [ …2] ] #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 {#1381 #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" => 1108001 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "87" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-285" "field_display_title" => "285" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1382 …2} "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 {#1383 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1384 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1385 …2} "content" => Illuminate\Database\Eloquent\Collection {#1386 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1393 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108001 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "87" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-285" "field_display_title" => "285" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1382 …2} "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 {#1383 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1384 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1385 …2} "content" => Illuminate\Database\Eloquent\Collection {#1386 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1393 …2} "page" => array:2 [ …2] ] #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 {#1394 #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" => 1108002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "87" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-286" "field_display_title" => "286" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1395 …2} "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 {#1396 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1397 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1398 …2} "content" => Illuminate\Database\Eloquent\Collection {#1399 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1406 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "87" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-286" "field_display_title" => "286" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1395 …2} "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 {#1396 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1397 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1398 …2} "content" => Illuminate\Database\Eloquent\Collection {#1399 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1406 …2} "page" => array:2 [ …2] ] #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 {#1407 #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" => 1108003 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "87" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-287" "field_display_title" => "287" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1408 …2} "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 {#1409 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1410 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1411 …2} "content" => Illuminate\Database\Eloquent\Collection {#1412 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1419 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108003 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "87" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-287" "field_display_title" => "287" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1408 …2} "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 {#1409 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1410 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1411 …2} "content" => Illuminate\Database\Eloquent\Collection {#1412 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1419 …2} "page" => array:2 [ …2] ] #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 {#1420 #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" => 1108004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "87" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-288" "field_display_title" => "288" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1421 …2} "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 {#1422 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1423 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1424 …2} "content" => Illuminate\Database\Eloquent\Collection {#1425 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1432 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "87" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-288" "field_display_title" => "288" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1421 …2} "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 {#1422 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1423 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1424 …2} "content" => Illuminate\Database\Eloquent\Collection {#1425 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1432 …2} "page" => array:2 [ …2] ] #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 {#1433 #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" => 1108005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "87" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-289" "field_display_title" => "289" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1434 …2} "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 {#1435 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1436 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1437 …2} "content" => Illuminate\Database\Eloquent\Collection {#1438 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1445 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "87" …20 ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 6 => App\Models\Task {#1446 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 7 => App\Models\Task {#1459 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 8 => App\Models\Task {#1472 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1097829 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "87" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-87" "field_display_title" => "87" "field_folder" => "folder1" "field_image_name" => "87" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1187} "field_weight" => "0" "field_book_parent" => 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№285 (с. 87)
Условие. №285 (с. 87)
скриншот условия
285. Вычислите:
а) $(\sqrt{5} + \sqrt{2})^5$;
б) $(\sqrt{3} - 1)^6$.
Решение. №285 (с. 87)
Решение 2 (rus). №285 (с. 87)
а) Для вычисления $(\sqrt{5} + \sqrt{2})^5$ воспользуемся формулой бинома Ньютона:
$(a+b)^n = C_n^0 a^n + C_n^1 a^{n-1}b + C_n^2 a^{n-2}b^2 + \dots + C_n^n b^n$
В нашем случае $a = \sqrt{5}$, $b = \sqrt{2}$, $n=5$. Биномиальные коэффициенты для $n=5$ равны: $C_5^0=1$, $C_5^1=5$, $C_5^2=10$, $C_5^3=10$, $C_5^4=5$, $C_5^5=1$.
Разложим выражение по формуле:
$(\sqrt{5} + \sqrt{2})^5 = C_5^0(\sqrt{5})^5 + C_5^1(\sqrt{5})^4(\sqrt{2})^1 + C_5^2(\sqrt{5})^3(\sqrt{2})^2 + C_5^3(\sqrt{5})^2(\sqrt{2})^3 + C_5^4(\sqrt{5})^1(\sqrt{2})^4 + C_5^5(\sqrt{2})^5$
Теперь вычислим каждый член разложения:
$C_5^0(\sqrt{5})^5 = 1 \cdot (\sqrt{5})^5 = (\sqrt{5}^2)^2 \cdot \sqrt{5} = 5^2 \cdot \sqrt{5} = 25\sqrt{5}$
$C_5^1(\sqrt{5})^4(\sqrt{2})^1 = 5 \cdot (\sqrt{5})^4 \cdot \sqrt{2} = 5 \cdot 5^2 \cdot \sqrt{2} = 5 \cdot 25 \cdot \sqrt{2} = 125\sqrt{2}$
$C_5^2(\sqrt{5})^3(\sqrt{2})^2 = 10 \cdot (5\sqrt{5}) \cdot 2 = 100\sqrt{5}$
$C_5^3(\sqrt{5})^2(\sqrt{2})^3 = 10 \cdot 5 \cdot (2\sqrt{2}) = 100\sqrt{2}$
$C_5^4(\sqrt{5})^1(\sqrt{2})^4 = 5 \cdot \sqrt{5} \cdot 4 = 20\sqrt{5}$
$C_5^5(\sqrt{2})^5 = 1 \cdot (\sqrt{2})^5 = (\sqrt{2}^2)^2 \cdot \sqrt{2} = 2^2 \cdot \sqrt{2} = 4\sqrt{2}$
Сложим все полученные значения:
$(\sqrt{5} + \sqrt{2})^5 = 25\sqrt{5} + 125\sqrt{2} + 100\sqrt{5} + 100\sqrt{2} + 20\sqrt{5} + 4\sqrt{2}$
Сгруппируем слагаемые с $\sqrt{5}$ и с $\sqrt{2}$:
$(25\sqrt{5} + 100\sqrt{5} + 20\sqrt{5}) + (125\sqrt{2} + 100\sqrt{2} + 4\sqrt{2}) = (25+100+20)\sqrt{5} + (125+100+4)\sqrt{2} = 145\sqrt{5} + 229\sqrt{2}$
Ответ: $145\sqrt{5} + 229\sqrt{2}$
б) Для вычисления $(\sqrt{3} - 1)^6$ также воспользуемся формулой бинома Ньютона для разности:
$(a-b)^n = C_n^0 a^n - C_n^1 a^{n-1}b + C_n^2 a^{n-2}b^2 - \dots + (-1)^n C_n^n b^n$
В данном случае $a = \sqrt{3}$, $b = 1$, $n=6$. Биномиальные коэффициенты для $n=6$ равны: $C_6^0=1$, $C_6^1=6$, $C_6^2=15$, $C_6^3=20$, $C_6^4=15$, $C_6^5=6$, $C_6^6=1$.
Разложим выражение по формуле, учитывая чередование знаков:
$(\sqrt{3} - 1)^6 = C_6^0(\sqrt{3})^6 - C_6^1(\sqrt{3})^5 \cdot 1^1 + C_6^2(\sqrt{3})^4 \cdot 1^2 - C_6^3(\sqrt{3})^3 \cdot 1^3 + C_6^4(\sqrt{3})^2 \cdot 1^4 - C_6^5(\sqrt{3})^1 \cdot 1^5 + C_6^6(\sqrt{3})^0 \cdot 1^6$
Вычислим каждый член разложения:
$C_6^0(\sqrt{3})^6 = 1 \cdot (\sqrt{3})^6 = 3^3 = 27$
$-C_6^1(\sqrt{3})^5 = -6 \cdot (\sqrt{3})^5 = -6 \cdot (3^2 \cdot \sqrt{3}) = -6 \cdot 9\sqrt{3} = -54\sqrt{3}$
$C_6^2(\sqrt{3})^4 = 15 \cdot (\sqrt{3})^4 = 15 \cdot 3^2 = 15 \cdot 9 = 135$
$-C_6^3(\sqrt{3})^3 = -20 \cdot (\sqrt{3})^3 = -20 \cdot (3\sqrt{3}) = -60\sqrt{3}$
$C_6^4(\sqrt{3})^2 = 15 \cdot (\sqrt{3})^2 = 15 \cdot 3 = 45$
$-C_6^5(\sqrt{3})^1 = -6 \cdot \sqrt{3} = -6\sqrt{3}$
$C_6^6(\sqrt{3})^0 = 1 \cdot (\sqrt{3})^0 \cdot 1^6 = 1 \cdot 1 \cdot 1 = 1$
Сложим все полученные значения:
$(\sqrt{3} - 1)^6 = 27 - 54\sqrt{3} + 135 - 60\sqrt{3} + 45 - 6\sqrt{3} + 1$
Сгруппируем рациональные и иррациональные слагаемые:
$(27 + 135 + 45 + 1) + (-54\sqrt{3} - 60\sqrt{3} - 6\sqrt{3}) = 208 - (54+60+6)\sqrt{3} = 208 - 120\sqrt{3}$
Ответ: $208 - 120\sqrt{3}$
Другие задания:
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