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Номер 46, страница 16 - 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
Повторение курса алгебры 8 класса. Упражнения - номер 46, страница 16.
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" => 1107736 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "16" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-46" "field_display_title" => "46" "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" => 1107636 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Повторение курса алгебры 8 класса" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "7" "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" => 1107636 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Повторение курса алгебры 8 класса" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333} "field_page_start" => "7" "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} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1060} ] #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 {#1061 #items: array:2 [ 0 => App\Models\Branch {#1023} 1 => App\Models\Branch {#1062 #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" => 1107637 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Упражнения" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "11" "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" => 1107636 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Повторение курса алгебры 8 класса" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_page_start" => "7" "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 {#1097 …2} ] #original: array:24 [ "id" => 1107636 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Повторение курса алгебры 8 класса" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_page_start" => "7" "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 {#1097 …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" => 1107637 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Упражнения" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063} "field_page_start" => "11" "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 {#1098 #items: array:2 [ 0 => App\Models\Element {#1099 #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" => 1277565 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100 #items: array:1 [ 0 => App\Models\Edition {#1101 #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 {#1102 …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 {#1104 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1106 …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 {#1102 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" 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"path" => "/media/algebra_09/soltan-u19/1-1/46-1.webp?ts=1752830738" ] 1 => array:5 [ "name" => "46-2.jpg" "alt" => null "width" => "1453" "height" => 901 "path" => "/media/algebra_09/soltan-u19/1-1/46-2.webp?ts=1752830738" ] ] ] #original: array:6 [ "id" => 1277565 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100} "task" => array:2 [ "refs" => "1107736" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "46-1.jpg" "alt" => null "width" => "1295" "height" => 697 "path" => "/media/algebra_09/soltan-u19/1-1/46-1.webp?ts=1752830738" ] 1 => array:5 [ "name" => "46-2.jpg" "alt" => null "width" => "1453" "height" => 901 "path" => "/media/algebra_09/soltan-u19/1-1/46-2.webp?ts=1752830738" ] ] ] #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 {#1107 #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" => 1553647 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:1 [ 0 => App\Models\Edition {#1109 #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 {#1110 …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 {#1112 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1114 …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 {#1110 …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 {#1112 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1114 …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" => "1107736" "type" => "task" ] "text" => "<p><strong>а)</strong> Для упрощения выражений вида $\sqrt{A \pm 2\sqrt{B}}$ используется формула выделения полного квадрата под корнем, основанная на формуле $(a \pm b)^2 = a^2 \pm 2ab + b^2$. Мы ищем такие $a$ и $b$, что $a^2+b^2=A$ и $ab=\sqrt{B}$.<br>Рассмотрим первое слагаемое $\sqrt{6+2\sqrt{5}}$. Здесь подкоренное выражение имеет вид $A+2\sqrt{B}$, где $A=6, B=5$. Нам нужно найти числа $a$ и $b$, такие что $a^2+b^2=6$ и $ab=\sqrt{5}$. Легко подобрать $a=\sqrt{5}$ и $b=1$.<br>Проверим: $a^2+b^2 = (\sqrt{5})^2+1^2 = 5+1=6$. Это верно.<br>Тогда $6+2\sqrt{5} = (\sqrt{5})^2 + 2 \cdot \sqrt{5} \cdot 1 + 1^2 = (\sqrt{5}+1)^2$.<br>Следовательно, $\sqrt{6+2\sqrt{5}} = \sqrt{(\sqrt{5}+1)^2} = |\sqrt{5}+1| = \sqrt{5}+1$.<br>Аналогично для второго слагаемого $\sqrt{6-2\sqrt{5}}$:<br>$6-2\sqrt{5} = (\sqrt{5})^2 - 2 \cdot \sqrt{5} \cdot 1 + 1^2 = (\sqrt{5}-1)^2$.<br>Следовательно, $\sqrt{6-2\sqrt{5}} = \sqrt{(\sqrt{5}-1)^2} = |\sqrt{5}-1| = \sqrt{5}-1$ (так как $\sqrt{5} \approx 2.23 > 1$).<br>Теперь вычисляем разность:<br>$\sqrt{6+2\sqrt{5}} - \sqrt{6-2\sqrt{5}} = (\sqrt{5}+1) - (\sqrt{5}-1) = \sqrt{5}+1-\sqrt{5}+1 = 2$.<br>Ответ: 2</p><p><strong>б)</strong> Преобразуем подкоренные выражения к виду $A \pm 2\sqrt{B}$, чтобы выделить полный квадрат.<br>$\sqrt{11-4\sqrt{7}} = \sqrt{11-2 \cdot 2\sqrt{7}} = \sqrt{11-2\sqrt{4 \cdot 7}} = \sqrt{11-2\sqrt{28}}$.<br>Ищем $a$ и $b$ такие, что $a^2+b^2=11$ и $ab=\sqrt{28}$. Число 28 можно разложить на множители 7 и 4. Попробуем $a=\sqrt{7}$ и $b=\sqrt{4}=2$.<br>Проверим: $a^2+b^2 = (\sqrt{7})^2+2^2 = 7+4=11$. Это верно.<br>Тогда $11-2\sqrt{28} = (\sqrt{7})^2 - 2 \cdot \sqrt{7} \cdot 2 + 2^2 = (\sqrt{7}-2)^2$.<br>Следовательно, $\sqrt{11-4\sqrt{7}} = \sqrt{(\sqrt{7}-2)^2} = |\sqrt{7}-2| = \sqrt{7}-2$ (так как $\sqrt{7} \approx 2.65 > 2$).<br>Аналогично для второго слагаемого:<br>$\sqrt{11+4\sqrt{7}} = \sqrt{11+2\sqrt{28}} = \sqrt{(\sqrt{7}+2)^2} = |\sqrt{7}+2| = \sqrt{7}+2$.<br>Теперь вычисляем сумму:<br>$\sqrt{11-4\sqrt{7}} + \sqrt{11+4\sqrt{7}} = (\sqrt{7}-2) + (\sqrt{7}+2) = \sqrt{7}-2+\sqrt{7}+2 = 2\sqrt{7}$.<br>Ответ: $2\sqrt{7}$</p><p><strong>в)</strong> Упростим выражение по частям.<br>Первая часть: $(1-2\sqrt{3})^2$. Используем формулу квадрата разности $(a-b)^2=a^2-2ab+b^2$.<br>$(1-2\sqrt{3})^2 = 1^2 - 2 \cdot 1 \cdot 2\sqrt{3} + (2\sqrt{3})^2 = 1 - 4\sqrt{3} + 4 \cdot 3 = 1 - 4\sqrt{3} + 12 = 13 - 4\sqrt{3}$.<br>Вторая часть: $4\sqrt{28+10\sqrt{3}}$. Упростим подкоренное выражение, приведя его к виду $A+2\sqrt{B}$.<br>$\sqrt{28+10\sqrt{3}} = \sqrt{28+2 \cdot 5\sqrt{3}} = \sqrt{28+2\sqrt{25 \cdot 3}} = \sqrt{28+2\sqrt{75}}$.<br>Ищем $a$ и $b$ такие, что $a^2+b^2=28$ и $ab=\sqrt{75}$. Число 75 можно разложить на множители 25 и 3. Попробуем $a=\sqrt{25}=5$ и $b=\sqrt{3}$.<br>Проверим: $a^2+b^2 = 5^2+(\sqrt{3})^2 = 25+3=28$. Это верно.<br>Тогда $28+2\sqrt{75} = (5+\sqrt{3})^2$.<br>Значит, $\sqrt{28+10\sqrt{3}} = \sqrt{(5+\sqrt{3})^2} = 5+\sqrt{3}$.<br>Вся вторая часть выражения равна $4(5+\sqrt{3}) = 20+4\sqrt{3}$.<br>Сложим обе упрощенные части:<br>$(13 - 4\sqrt{3}) + (20+4\sqrt{3}) = 13 - 4\sqrt{3} + 20 + 4\sqrt{3} = 33$.<br>Ответ: 33</p><p><strong>г)</strong> Упростим выражение по частям.<br>Первая часть: $(3\sqrt{2}+5)^2$. Используем формулу квадрата суммы $(a+b)^2=a^2+2ab+b^2$.<br>$(3\sqrt{2}+5)^2 = (3\sqrt{2})^2 + 2 \cdot 3\sqrt{2} \cdot 5 + 5^2 = 9 \cdot 2 + 30\sqrt{2} + 25 = 18 + 30\sqrt{2} + 25 = 43 + 30\sqrt{2}$.<br>Вторая часть: $30\sqrt{51-14\sqrt{2}}$. Упростим подкоренное выражение, приведя его к виду $A-2\sqrt{B}$.<br>$\sqrt{51-14\sqrt{2}} = \sqrt{51-2 \cdot 7\sqrt{2}} = \sqrt{51-2\sqrt{49 \cdot 2}} = \sqrt{51-2\sqrt{98}}$.<br>Ищем $a$ и $b$ такие, что $a^2+b^2=51$ и $ab=\sqrt{98}$. Число 98 можно разложить на множители 49 и 2. Попробуем $a=\sqrt{49}=7$ и $b=\sqrt{2}$.<br>Проверим: $a^2+b^2=7^2+(\sqrt{2})^2=49+2=51$. Это верно.<br>Тогда $51-2\sqrt{98} = (7-\sqrt{2})^2$.<br>Значит, $\sqrt{51-14\sqrt{2}} = \sqrt{(7-\sqrt{2})^2} = |7-\sqrt{2}| = 7-\sqrt{2}$ (так как $7 > \sqrt{2}$).<br>Вся вторая часть выражения равна $30(7-\sqrt{2}) = 210-30\sqrt{2}$.<br>Сложим обе упрощенные части:<br>$(43 + 30\sqrt{2}) + (210-30\sqrt{2}) = 43 + 30\sqrt{2} + 210 - 30\sqrt{2} = 253$.<br>Ответ: 253</p>" ] #original: array:6 [ "id" => 1553647 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108} "task" => array:2 [ "refs" => "1107736" "type" => "task" ] "text" => "<p><strong>а)</strong> Для упрощения выражений вида $\sqrt{A \pm 2\sqrt{B}}$ используется формула выделения полного квадрата под корнем, основанная на формуле $(a \pm b)^2 = a^2 \pm 2ab + b^2$. Мы ищем такие $a$ и $b$, что $a^2+b^2=A$ и $ab=\sqrt{B}$.<br>Рассмотрим первое слагаемое $\sqrt{6+2\sqrt{5}}$. Здесь подкоренное выражение имеет вид $A+2\sqrt{B}$, где $A=6, B=5$. Нам нужно найти числа $a$ и $b$, такие что $a^2+b^2=6$ и $ab=\sqrt{5}$. Легко подобрать $a=\sqrt{5}$ и $b=1$.<br>Проверим: $a^2+b^2 = (\sqrt{5})^2+1^2 = 5+1=6$. Это верно.<br>Тогда $6+2\sqrt{5} = (\sqrt{5})^2 + 2 \cdot \sqrt{5} \cdot 1 + 1^2 = (\sqrt{5}+1)^2$.<br>Следовательно, $\sqrt{6+2\sqrt{5}} = \sqrt{(\sqrt{5}+1)^2} = |\sqrt{5}+1| = \sqrt{5}+1$.<br>Аналогично для второго слагаемого $\sqrt{6-2\sqrt{5}}$:<br>$6-2\sqrt{5} = (\sqrt{5})^2 - 2 \cdot \sqrt{5} \cdot 1 + 1^2 = (\sqrt{5}-1)^2$.<br>Следовательно, $\sqrt{6-2\sqrt{5}} = \sqrt{(\sqrt{5}-1)^2} = |\sqrt{5}-1| = \sqrt{5}-1$ (так как $\sqrt{5} \approx 2.23 > 1$).<br>Теперь вычисляем разность:<br>$\sqrt{6+2\sqrt{5}} - \sqrt{6-2\sqrt{5}} = (\sqrt{5}+1) - (\sqrt{5}-1) = \sqrt{5}+1-\sqrt{5}+1 = 2$.<br>Ответ: 2</p><p><strong>б)</strong> Преобразуем подкоренные выражения к виду $A \pm 2\sqrt{B}$, чтобы выделить полный квадрат.<br>$\sqrt{11-4\sqrt{7}} = \sqrt{11-2 \cdot 2\sqrt{7}} = \sqrt{11-2\sqrt{4 \cdot 7}} = \sqrt{11-2\sqrt{28}}$.<br>Ищем $a$ и $b$ такие, что $a^2+b^2=11$ и $ab=\sqrt{28}$. Число 28 можно разложить на множители 7 и 4. Попробуем $a=\sqrt{7}$ и $b=\sqrt{4}=2$.<br>Проверим: $a^2+b^2 = (\sqrt{7})^2+2^2 = 7+4=11$. Это верно.<br>Тогда $11-2\sqrt{28} = (\sqrt{7})^2 - 2 \cdot \sqrt{7} \cdot 2 + 2^2 = (\sqrt{7}-2)^2$.<br>Следовательно, $\sqrt{11-4\sqrt{7}} = \sqrt{(\sqrt{7}-2)^2} = |\sqrt{7}-2| = \sqrt{7}-2$ (так как $\sqrt{7} \approx 2.65 > 2$).<br>Аналогично для второго слагаемого:<br>$\sqrt{11+4\sqrt{7}} = \sqrt{11+2\sqrt{28}} = \sqrt{(\sqrt{7}+2)^2} = |\sqrt{7}+2| = \sqrt{7}+2$.<br>Теперь вычисляем сумму:<br>$\sqrt{11-4\sqrt{7}} + \sqrt{11+4\sqrt{7}} = (\sqrt{7}-2) + (\sqrt{7}+2) = \sqrt{7}-2+\sqrt{7}+2 = 2\sqrt{7}$.<br>Ответ: $2\sqrt{7}$</p><p><strong>в)</strong> Упростим выражение по частям.<br>Первая часть: $(1-2\sqrt{3})^2$. Используем формулу квадрата разности $(a-b)^2=a^2-2ab+b^2$.<br>$(1-2\sqrt{3})^2 = 1^2 - 2 \cdot 1 \cdot 2\sqrt{3} + (2\sqrt{3})^2 = 1 - 4\sqrt{3} + 4 \cdot 3 = 1 - 4\sqrt{3} + 12 = 13 - 4\sqrt{3}$.<br>Вторая часть: $4\sqrt{28+10\sqrt{3}}$. Упростим подкоренное выражение, приведя его к виду $A+2\sqrt{B}$.<br>$\sqrt{28+10\sqrt{3}} = \sqrt{28+2 \cdot 5\sqrt{3}} = \sqrt{28+2\sqrt{25 \cdot 3}} = \sqrt{28+2\sqrt{75}}$.<br>Ищем $a$ и $b$ такие, что $a^2+b^2=28$ и $ab=\sqrt{75}$. Число 75 можно разложить на множители 25 и 3. Попробуем $a=\sqrt{25}=5$ и $b=\sqrt{3}$.<br>Проверим: $a^2+b^2 = 5^2+(\sqrt{3})^2 = 25+3=28$. Это верно.<br>Тогда $28+2\sqrt{75} = (5+\sqrt{3})^2$.<br>Значит, $\sqrt{28+10\sqrt{3}} = \sqrt{(5+\sqrt{3})^2} = 5+\sqrt{3}$.<br>Вся вторая часть выражения равна $4(5+\sqrt{3}) = 20+4\sqrt{3}$.<br>Сложим обе упрощенные части:<br>$(13 - 4\sqrt{3}) + (20+4\sqrt{3}) = 13 - 4\sqrt{3} + 20 + 4\sqrt{3} = 33$.<br>Ответ: 33</p><p><strong>г)</strong> Упростим выражение по частям.<br>Первая часть: $(3\sqrt{2}+5)^2$. Используем формулу квадрата суммы $(a+b)^2=a^2+2ab+b^2$.<br>$(3\sqrt{2}+5)^2 = (3\sqrt{2})^2 + 2 \cdot 3\sqrt{2} \cdot 5 + 5^2 = 9 \cdot 2 + 30\sqrt{2} + 25 = 18 + 30\sqrt{2} + 25 = 43 + 30\sqrt{2}$.<br>Вторая часть: $30\sqrt{51-14\sqrt{2}}$. Упростим подкоренное выражение, приведя его к виду $A-2\sqrt{B}$.<br>$\sqrt{51-14\sqrt{2}} = \sqrt{51-2 \cdot 7\sqrt{2}} = \sqrt{51-2\sqrt{49 \cdot 2}} = \sqrt{51-2\sqrt{98}}$.<br>Ищем $a$ и $b$ такие, что $a^2+b^2=51$ и $ab=\sqrt{98}$. Число 98 можно разложить на множители 49 и 2. Попробуем $a=\sqrt{49}=7$ и $b=\sqrt{2}$.<br>Проверим: $a^2+b^2=7^2+(\sqrt{2})^2=49+2=51$. Это верно.<br>Тогда $51-2\sqrt{98} = (7-\sqrt{2})^2$.<br>Значит, $\sqrt{51-14\sqrt{2}} = \sqrt{(7-\sqrt{2})^2} = |7-\sqrt{2}| = 7-\sqrt{2}$ (так как $7 > \sqrt{2}$).<br>Вся вторая часть выражения равна $30(7-\sqrt{2}) = 210-30\sqrt{2}$.<br>Сложим обе упрощенные части:<br>$(43 + 30\sqrt{2}) + (210-30\sqrt{2}) = 43 + 30\sqrt{2} + 210 - 30\sqrt{2} = 253$.<br>Ответ: 253</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" => "1107737" "type" => "task" ] "previous" => array:2 [ "refs" => "1107735" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1115 #items: array:1 [ 0 => App\Models\Book {#1116 #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 {#1117 #items: array:1 [ 0 => App\Models\Term {#1118 #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 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: 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#classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Term {#1126 #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" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" 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"/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1179 …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 {#1153 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1155 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1157 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1159 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1164 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1166 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1168 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1170 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1172 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1174 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1175 …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 {#1176 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1177 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1178 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1179 …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 {#1180 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1181 #items: array:1 [ 0 => App\Models\Element {#1182 #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" => 1260909 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1183 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1260909 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1183 …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" => "1097759" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097757" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1190 #items: array:9 [ 0 => App\Models\Task {#1191 #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" => 1107732 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "16" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-42" "field_display_title" => "42" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1192 …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 {#1194 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1195 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1200 …2} "content" => Illuminate\Database\Eloquent\Collection {#1237 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1256 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107732 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "16" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-42" "field_display_title" => "42" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1192 …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 {#1194 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1195 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1200 …2} "content" => Illuminate\Database\Eloquent\Collection {#1237 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1256 …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 {#1257 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107733 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "16" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-43" "field_display_title" => "43" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1258 …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 {#1259 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1260 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1261 …2} "content" => Illuminate\Database\Eloquent\Collection {#1262 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1269 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107733 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "16" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-43" "field_display_title" => "43" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1258 …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 {#1259 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1260 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1261 …2} "content" => Illuminate\Database\Eloquent\Collection {#1262 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1269 …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 {#1270 #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" => 1107734 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "16" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-44" "field_display_title" => "44" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1271 …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 {#1272 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1273 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1274 …2} "content" => Illuminate\Database\Eloquent\Collection {#1275 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1280 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107734 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "16" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-44" "field_display_title" => "44" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1271 …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 {#1272 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1273 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1274 …2} "content" => Illuminate\Database\Eloquent\Collection {#1275 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1280 …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 {#1281 #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" => 1107735 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "16" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-45" "field_display_title" => "45" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1282 …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 {#1283 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1284 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1285 …2} "content" => Illuminate\Database\Eloquent\Collection {#1286 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1291 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107735 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "16" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-45" "field_display_title" => "45" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1282 …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 {#1283 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1284 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1285 …2} "content" => Illuminate\Database\Eloquent\Collection {#1286 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1291 …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 {#1292 #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" => 1107736 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "16" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-46" "field_display_title" => "46" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1293 …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 {#1294 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1295 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1296 …2} "content" => Illuminate\Database\Eloquent\Collection {#1297 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1302 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107736 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "16" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-46" "field_display_title" => "46" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1293 …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 {#1294 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1295 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1296 …2} "content" => Illuminate\Database\Eloquent\Collection {#1297 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1302 …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 {#1303 #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" => 1107737 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "16" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-47" "field_display_title" => "47" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1304 …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 {#1305 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1306 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1307 …2} "content" => Illuminate\Database\Eloquent\Collection {#1308 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1313 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107737 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "16" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-47" "field_display_title" => "47" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1304 …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 {#1305 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1306 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1307 …2} "content" => Illuminate\Database\Eloquent\Collection {#1308 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1313 …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 => "*" ] } 6 => App\Models\Task {#1314 #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" => 1107738 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "16" 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[] #guarded: array:1 [ 0 => "*" ] }
№46 (с. 16)
Решение. №46 (с. 16)
Решение 2 (rus). №46 (с. 16)
а) Для упрощения выражений вида $\sqrt{A \pm 2\sqrt{B}}$ используется формула выделения полного квадрата под корнем, основанная на формуле $(a \pm b)^2 = a^2 \pm 2ab + b^2$. Мы ищем такие $a$ и $b$, что $a^2+b^2=A$ и $ab=\sqrt{B}$.
Рассмотрим первое слагаемое $\sqrt{6+2\sqrt{5}}$. Здесь подкоренное выражение имеет вид $A+2\sqrt{B}$, где $A=6, B=5$. Нам нужно найти числа $a$ и $b$, такие что $a^2+b^2=6$ и $ab=\sqrt{5}$. Легко подобрать $a=\sqrt{5}$ и $b=1$.
Проверим: $a^2+b^2 = (\sqrt{5})^2+1^2 = 5+1=6$. Это верно.
Тогда $6+2\sqrt{5} = (\sqrt{5})^2 + 2 \cdot \sqrt{5} \cdot 1 + 1^2 = (\sqrt{5}+1)^2$.
Следовательно, $\sqrt{6+2\sqrt{5}} = \sqrt{(\sqrt{5}+1)^2} = |\sqrt{5}+1| = \sqrt{5}+1$.
Аналогично для второго слагаемого $\sqrt{6-2\sqrt{5}}$:
$6-2\sqrt{5} = (\sqrt{5})^2 - 2 \cdot \sqrt{5} \cdot 1 + 1^2 = (\sqrt{5}-1)^2$.
Следовательно, $\sqrt{6-2\sqrt{5}} = \sqrt{(\sqrt{5}-1)^2} = |\sqrt{5}-1| = \sqrt{5}-1$ (так как $\sqrt{5} \approx 2.23 > 1$).
Теперь вычисляем разность:
$\sqrt{6+2\sqrt{5}} - \sqrt{6-2\sqrt{5}} = (\sqrt{5}+1) - (\sqrt{5}-1) = \sqrt{5}+1-\sqrt{5}+1 = 2$.
Ответ: 2
б) Преобразуем подкоренные выражения к виду $A \pm 2\sqrt{B}$, чтобы выделить полный квадрат.
$\sqrt{11-4\sqrt{7}} = \sqrt{11-2 \cdot 2\sqrt{7}} = \sqrt{11-2\sqrt{4 \cdot 7}} = \sqrt{11-2\sqrt{28}}$.
Ищем $a$ и $b$ такие, что $a^2+b^2=11$ и $ab=\sqrt{28}$. Число 28 можно разложить на множители 7 и 4. Попробуем $a=\sqrt{7}$ и $b=\sqrt{4}=2$.
Проверим: $a^2+b^2 = (\sqrt{7})^2+2^2 = 7+4=11$. Это верно.
Тогда $11-2\sqrt{28} = (\sqrt{7})^2 - 2 \cdot \sqrt{7} \cdot 2 + 2^2 = (\sqrt{7}-2)^2$.
Следовательно, $\sqrt{11-4\sqrt{7}} = \sqrt{(\sqrt{7}-2)^2} = |\sqrt{7}-2| = \sqrt{7}-2$ (так как $\sqrt{7} \approx 2.65 > 2$).
Аналогично для второго слагаемого:
$\sqrt{11+4\sqrt{7}} = \sqrt{11+2\sqrt{28}} = \sqrt{(\sqrt{7}+2)^2} = |\sqrt{7}+2| = \sqrt{7}+2$.
Теперь вычисляем сумму:
$\sqrt{11-4\sqrt{7}} + \sqrt{11+4\sqrt{7}} = (\sqrt{7}-2) + (\sqrt{7}+2) = \sqrt{7}-2+\sqrt{7}+2 = 2\sqrt{7}$.
Ответ: $2\sqrt{7}$
в) Упростим выражение по частям.
Первая часть: $(1-2\sqrt{3})^2$. Используем формулу квадрата разности $(a-b)^2=a^2-2ab+b^2$.
$(1-2\sqrt{3})^2 = 1^2 - 2 \cdot 1 \cdot 2\sqrt{3} + (2\sqrt{3})^2 = 1 - 4\sqrt{3} + 4 \cdot 3 = 1 - 4\sqrt{3} + 12 = 13 - 4\sqrt{3}$.
Вторая часть: $4\sqrt{28+10\sqrt{3}}$. Упростим подкоренное выражение, приведя его к виду $A+2\sqrt{B}$.
$\sqrt{28+10\sqrt{3}} = \sqrt{28+2 \cdot 5\sqrt{3}} = \sqrt{28+2\sqrt{25 \cdot 3}} = \sqrt{28+2\sqrt{75}}$.
Ищем $a$ и $b$ такие, что $a^2+b^2=28$ и $ab=\sqrt{75}$. Число 75 можно разложить на множители 25 и 3. Попробуем $a=\sqrt{25}=5$ и $b=\sqrt{3}$.
Проверим: $a^2+b^2 = 5^2+(\sqrt{3})^2 = 25+3=28$. Это верно.
Тогда $28+2\sqrt{75} = (5+\sqrt{3})^2$.
Значит, $\sqrt{28+10\sqrt{3}} = \sqrt{(5+\sqrt{3})^2} = 5+\sqrt{3}$.
Вся вторая часть выражения равна $4(5+\sqrt{3}) = 20+4\sqrt{3}$.
Сложим обе упрощенные части:
$(13 - 4\sqrt{3}) + (20+4\sqrt{3}) = 13 - 4\sqrt{3} + 20 + 4\sqrt{3} = 33$.
Ответ: 33
г) Упростим выражение по частям.
Первая часть: $(3\sqrt{2}+5)^2$. Используем формулу квадрата суммы $(a+b)^2=a^2+2ab+b^2$.
$(3\sqrt{2}+5)^2 = (3\sqrt{2})^2 + 2 \cdot 3\sqrt{2} \cdot 5 + 5^2 = 9 \cdot 2 + 30\sqrt{2} + 25 = 18 + 30\sqrt{2} + 25 = 43 + 30\sqrt{2}$.
Вторая часть: $30\sqrt{51-14\sqrt{2}}$. Упростим подкоренное выражение, приведя его к виду $A-2\sqrt{B}$.
$\sqrt{51-14\sqrt{2}} = \sqrt{51-2 \cdot 7\sqrt{2}} = \sqrt{51-2\sqrt{49 \cdot 2}} = \sqrt{51-2\sqrt{98}}$.
Ищем $a$ и $b$ такие, что $a^2+b^2=51$ и $ab=\sqrt{98}$. Число 98 можно разложить на множители 49 и 2. Попробуем $a=\sqrt{49}=7$ и $b=\sqrt{2}$.
Проверим: $a^2+b^2=7^2+(\sqrt{2})^2=49+2=51$. Это верно.
Тогда $51-2\sqrt{98} = (7-\sqrt{2})^2$.
Значит, $\sqrt{51-14\sqrt{2}} = \sqrt{(7-\sqrt{2})^2} = |7-\sqrt{2}| = 7-\sqrt{2}$ (так как $7 > \sqrt{2}$).
Вся вторая часть выражения равна $30(7-\sqrt{2}) = 210-30\sqrt{2}$.
Сложим обе упрощенные части:
$(43 + 30\sqrt{2}) + (210-30\sqrt{2}) = 43 + 30\sqrt{2} + 210 - 30\sqrt{2} = 253$.
Ответ: 253
Другие задания:
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