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Номер 8.7, страница 52, часть 2 - гдз по алгебре 8 класс учебник Мордкович, Александрова
Авторы: Мордкович А. Г., Александрова Л. А., Мишустина Т. Н., Тульчинская Е. Е., Семенов П. В.
Тип: Учебник
Издательство: Мнемозина
Год издания: 2019 - 2026
Часть: 2
Допущено Министерством просвещения Российской Федерации
Популярные ГДЗ в 8 классе
Часть 2. Глава 1. Алгебраические дроби. Параграф 8. Степень с отрицательным целым показателем - номер 8.7, страница 52.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 227164 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "52" "field_page_end" => null "field_url" => "/8-klass/algebra/mordkovich/08-7" "field_display_title" => "8.7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ 0 => App\Models\Branch {#1045 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 226557 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1050 #items: array:1 [ 0 => App\Models\Term {#1047 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #original: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "4" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => array:1 [ 0 => "/media/algebra_08/mordkovich-z-21/covers/cover2.webp?ts=1734210650" ] "field_branch_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_08/mordkovich-z-21/covers/cover2.webp?ts=1734210650" "alt" => "часть 2" "width" => "1875" "height" => "2548" ] ] "book" => Illuminate\Database\Eloquent\Collection {#1058 #items: array:1 [ 0 => App\Models\Book {#1057 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 14 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1062 #items: array:1 [ 0 => App\Models\Term {#1059 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1064 #items: array:1 [ 0 => App\Models\Term {#1061 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1066 #items: array:1 [ 0 => App\Models\Term {#1063 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1067 #items: array:5 [ 0 => App\Models\Term {#1080 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Term {#1078 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Term {#1079 #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 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 3 => App\Models\Term {#1082 #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 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 4 => App\Models\Term {#1081 #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 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1076 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1073 #items: array:1 [ 0 => App\Models\Term {#1060 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1074 #items: array:1 [ 0 => App\Models\Term {#1075 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1068 #items: array:1 [ 0 => App\Models\Term {#1072 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1070 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1065 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1069 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => "26" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1086 #items: array:1 [ 0 => App\Models\Term {#1083 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1085 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "2014" "field_priority" => "4" "field_default_folder" => "/algebra_08/mordkovich-z-21/" "field_isbn" => null "field_cover" => array:2 [ 0 => "/media/algebra_08/mordkovich-z-21/covers/cover1.webp?ts=1734210650" 1 => "/media/algebra_08/mordkovich-z-21/covers/cover2.webp?ts=1734210650" ] "field_cover_alts" => array:2 [ 0 => "часть 1" 1 => "часть 2" ] "field_covers" => array:2 [ 0 => array:4 [ "path" => "/media/algebra_08/mordkovich-z-21/covers/cover1.webp?ts=1734210650" "alt" => "часть 1" "width" => "1919" "height" => "2577" ] 1 => array:4 [ "path" => "/media/algebra_08/mordkovich-z-21/covers/cover2.webp?ts=1734210650" "alt" => "часть 2" "width" => "1875" "height" => "2548" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1084 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1087 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1088 #items: [] #escapeWhenCastingToString: false } "field_url" => "/8-klass/algebra/mordkovich" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1089 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => "<!--Текст в низу страницы--> <h2> Чем порадует учеников алгебра в 8 классе? </h2> <p> Школьники все надеются, что программа обучения станет немного более понятной и простой. Особенно это касается алгебры - предмета со всех сторон весьма непростого. Но в восьмом классе их надеждам тоже не суждено сбыться. Мало того, что материал предстоит учить весьма обширный, количество домашних заданий увеличится, так еще добавится подготовка к предстоящим вскоре ОГЭ. Чтобы справиться со всем этим, учащимся стоит воспользоваться «<strong>ГДЗ по Алгебре 8 класс Учебник Мордкович (Мнемозина, 2021 г.)</strong>». </p> <p> Школьникам в этом году предстоит изучать такие сложные темы как: <ol> <li>Действия с алгебраическими дробями.</li> <li>Преобразование рациональных уравнений.</li> <li>Теория вероятности.</li> <li>Модуль числа.</li> <li>Линейные и квадратные неравенства.</li> <li>Комбинаторность, и т.д.</li> </ol> <p> Больше всего трудностей возникает при построении графиков функций, которые перед этим нужно найти. Подобрать правильное уравнение весьма непросто, особенно если нет понимания, как именно это делается. Как показывает практика, многие ученики испытывают проблемы именно при решении неравенств. У подростков часто нет времени сидеть и детально изучать в чем именно прослеживается недочет. Этого можно сразу и не понять, если раньше были упущены какие-то навыки. Поэтому решебник станет отличным вариантом для устранения всех возникающих недопониманий. </p> <h2> Насколько полезен решебник к учебнику Мордковича? </h2> <p> Споры касательно применения электронных справочников ведутся уже давно. Но школьники обычно в них не участвуют, так как прекрасно понимают незаменимость и актуальность этих пособий. <strong>Готовые решения по алгебре за 8 класс к учебнику(задачнику) Мордковича А. Г.</strong> позволяют им полноценно освоить все аспекты учебного материала, восполнить пробелы и проработать проблемные моменты, не затрачивая на это много времени и сил. </p> <p> Конечно, достигнуть хороших результатов можно только при определенных условиях:</p> <ol> <li>никогда нельзя забывать про изучение теории, даже если ее в учебнике самый минимум;</li> <li>домашние задания всегда нужно выполнять самостоятельно, без подсказок;</li> <li>сверяя решения по сборнику, стоит помнить о внимательности;</li> <li>если найдены ошибки, их необходимо не только исправить, но и понять, почему они возникли.</li> </ol> <p> Только такой подход гарантирует успешное освоение дисциплины. Однако у учеников всегда присутствует соблазн просто списать ответы. Принесет ли им это пользу? Безусловно, пару-тройку хороших оценок они получат, но что дальше? Преподаватель рано или поздно заметит обман и придется нагонять программу, которая стремительно несется вперед. К тому же, доверие учителя уже будет потеряно, что может положить конец отличной успеваемости. Так что рисковать явно не стоит. </p> <h2> Решебник - уникальное учебное пособие </h2> <p> У каждого ребенка свои таланты. Одним хорошо даются точные науки, другим - гуманитарные. Но практически все школьники испытывают трудности при освоении алгебры. Лишь единицы подростков легко ориентируются в материале, формулах и уравнениях. Остальным же требуется дополнительная помощь. Оказать ее могут <strong>подробные решения по алгебре 8 класс Мордковича</strong>. </p> <p> В справочнике размещены все необходимые сведения, которые помогут усвоить изучаемый параграф: <ul> <li>верные ответы на все номера;</li> <li>доскональные решения;</li> <li>подробные наглядные примеры;</li> <li>дополнительные примечания.</li> </ul> <p> Сборник создан группой профессиональных методистов, которые максимально подробно описали все действия, что нужно совершить ученикам, чтобы получить правильные результаты. Периодически сверяясь с решебником восьмиклассники получают возможность удостовериться в своей правоте, либо проработать недочеты. </p>" "breadcrumbs" => array:3 [ "class" => 379 "subject" => 542 "class_subject" => 385 ] ] #original: array:50 [ "id" => 14 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1062} "field_class" => Illuminate\Database\Eloquent\Collection {#1064} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1066} "field_author" => Illuminate\Database\Eloquent\Collection {#1067} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1076} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1073} "field_country" => Illuminate\Database\Eloquent\Collection {#1074} "field_city" => Illuminate\Database\Eloquent\Collection {#1068} "field_series" => Illuminate\Database\Eloquent\Collection {#1070} "field_umk" => Illuminate\Database\Eloquent\Collection {#1065} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1071} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1069} "field_publication_number" => "26" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1086} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1085} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "2014" "field_priority" => "4" "field_default_folder" => "/algebra_08/mordkovich-z-21/" "field_isbn" => null "field_cover" => array:2 [ 0 => "/media/algebra_08/mordkovich-z-21/covers/cover1.webp?ts=1734210650" 1 => "/media/algebra_08/mordkovich-z-21/covers/cover2.webp?ts=1734210650" ] "field_cover_alts" => array:2 [ 0 => "часть 1" 1 => "часть 2" ] "field_covers" => array:2 [ 0 => array:4 [ "path" => "/media/algebra_08/mordkovich-z-21/covers/cover1.webp?ts=1734210650" "alt" => "часть 1" "width" => "1919" "height" => "2577" ] 1 => array:4 [ "path" => "/media/algebra_08/mordkovich-z-21/covers/cover2.webp?ts=1734210650" "alt" => "часть 2" "width" => "1875" "height" => "2548" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1084} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1087} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1088} "field_url" => "/8-klass/algebra/mordkovich" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1089} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => "<!--Текст в низу страницы--> <h2> Чем порадует учеников алгебра в 8 классе? </h2> <p> Школьники все надеются, что программа обучения станет немного более понятной и простой. Особенно это касается алгебры - предмета со всех сторон весьма непростого. Но в восьмом классе их надеждам тоже не суждено сбыться. Мало того, что материал предстоит учить весьма обширный, количество домашних заданий увеличится, так еще добавится подготовка к предстоящим вскоре ОГЭ. Чтобы справиться со всем этим, учащимся стоит воспользоваться «<strong>ГДЗ по Алгебре 8 класс Учебник Мордкович (Мнемозина, 2021 г.)</strong>». </p> <p> Школьникам в этом году предстоит изучать такие сложные темы как: <ol> <li>Действия с алгебраическими дробями.</li> <li>Преобразование рациональных уравнений.</li> <li>Теория вероятности.</li> <li>Модуль числа.</li> <li>Линейные и квадратные неравенства.</li> <li>Комбинаторность, и т.д.</li> </ol> <p> Больше всего трудностей возникает при построении графиков функций, которые перед этим нужно найти. Подобрать правильное уравнение весьма непросто, особенно если нет понимания, как именно это делается. Как показывает практика, многие ученики испытывают проблемы именно при решении неравенств. У подростков часто нет времени сидеть и детально изучать в чем именно прослеживается недочет. Этого можно сразу и не понять, если раньше были упущены какие-то навыки. Поэтому решебник станет отличным вариантом для устранения всех возникающих недопониманий. </p> <h2> Насколько полезен решебник к учебнику Мордковича? </h2> <p> Споры касательно применения электронных справочников ведутся уже давно. Но школьники обычно в них не участвуют, так как прекрасно понимают незаменимость и актуальность этих пособий. <strong>Готовые решения по алгебре за 8 класс к учебнику(задачнику) Мордковича А. Г.</strong> позволяют им полноценно освоить все аспекты учебного материала, восполнить пробелы и проработать проблемные моменты, не затрачивая на это много времени и сил. </p> <p> Конечно, достигнуть хороших результатов можно только при определенных условиях:</p> <ol> <li>никогда нельзя забывать про изучение теории, даже если ее в учебнике самый минимум;</li> <li>домашние задания всегда нужно выполнять самостоятельно, без подсказок;</li> <li>сверяя решения по сборнику, стоит помнить о внимательности;</li> <li>если найдены ошибки, их необходимо не только исправить, но и понять, почему они возникли.</li> </ol> <p> Только такой подход гарантирует успешное освоение дисциплины. Однако у учеников всегда присутствует соблазн просто списать ответы. Принесет ли им это пользу? Безусловно, пару-тройку хороших оценок они получат, но что дальше? Преподаватель рано или поздно заметит обман и придется нагонять программу, которая стремительно несется вперед. К тому же, доверие учителя уже будет потеряно, что может положить конец отличной успеваемости. Так что рисковать явно не стоит. </p> <h2> Решебник - уникальное учебное пособие </h2> <p> У каждого ребенка свои таланты. Одним хорошо даются точные науки, другим - гуманитарные. Но практически все школьники испытывают трудности при освоении алгебры. Лишь единицы подростков легко ориентируются в материале, формулах и уравнениях. Остальным же требуется дополнительная помощь. Оказать ее могут <strong>подробные решения по алгебре 8 класс Мордковича</strong>. </p> <p> В справочнике размещены все необходимые сведения, которые помогут усвоить изучаемый параграф: <ul> <li>верные ответы на все номера;</li> <li>доскональные решения;</li> <li>подробные наглядные примеры;</li> <li>дополнительные примечания.</li> </ul> <p> Сборник создан группой профессиональных методистов, которые максимально подробно описали все действия, что нужно совершить ученикам, чтобы получить правильные результаты. 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=> "Степень с отрицательным целым показателем" "field_branch_order" => "8" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1137 #items: array:1 [ 0 => App\Models\Term {#1134 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "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_page_start" => "52" "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 {#1136 #items: array:1 [ 0 => App\Models\Book {#1057} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1145 #items: array:1 [ 0 => App\Models\Branch {#1144 #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" => 226559 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Алгебраические дроби" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1146 …2} "field_page_start" => "12" "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 {#1147 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1148 …2} ] #original: array:24 [ "id" => 226559 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Алгебраические дроби" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1146 …2} "field_page_start" => "12" "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 {#1147 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1148 …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" => 226567 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Степень с отрицательным целым показателем" "field_branch_order" => "8" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1137} "field_page_start" => "52" "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 {#1136} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1145} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1038 #items: array:6 [ 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:7 [ "id" => 265811 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1051 #items: array:1 [ 0 => App\Models\Edition {#1043 #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" => 723 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1041 …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" => "vadim" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1118 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1115 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_content_source" => null ] #original: array:21 [ "id" => 723 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1041 …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" => "vadim" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1118 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1115 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1113 …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" => "227164" "type" => "task" ] "text" => "<p><strong>8.7</strong> Представьте числа 2, 4, 8, 16, 32, 64, 128, $\frac{1}{2}$, $\frac{1}{4}$, $\frac{1}{8}$, $\frac{1}{32}$, $\frac{1}{128}$ в виде степени числа:</p><p><strong>а)</strong> 2;</p><p><strong>б)</strong> $\frac{1}{2}$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "7-1.jpg" "alt" => null "width" => "1467" "height" => 330 "path" => "/media/algebra_08/mordkovich-z-21/0-2-08/7-1.webp?ts=1734378453" ] ] ] #original: array:7 [ "id" => 265811 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1051} "task" => array:2 [ "refs" => "227164" "type" => "task" ] "text" => "<p><strong>8.7</strong> Представьте числа 2, 4, 8, 16, 32, 64, 128, $\frac{1}{2}$, $\frac{1}{4}$, $\frac{1}{8}$, $\frac{1}{32}$, $\frac{1}{128}$ в виде степени числа:</p><p><strong>а)</strong> 2;</p><p><strong>б)</strong> $\frac{1}{2}$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "7-1.jpg" "alt" => null "width" => "1467" "height" => 330 "path" => "/media/algebra_08/mordkovich-z-21/0-2-08/7-1.webp?ts=1734378453" ] ] ] #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 {#1095 #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" => 273020 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1158 #items: array:1 [ 0 => App\Models\Edition {#1160 #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" => 724 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 1" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1171 …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" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1168 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1164 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1161 …2} "field_content_source" => null ] #original: array:21 [ "id" => 724 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 1" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1171 …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" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1168 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1164 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1161 …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" => "227164" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "7-1.png" "alt" => null "width" => "705" "height" => 1680 "path" => "/media/algebra_08/mordkovich-z-21/1-2-08/7-1.webp?ts=1734380238" ] 1 => array:5 [ "name" => "7-2.png" "alt" => null "width" => "705" "height" => 1977 "path" => "/media/algebra_08/mordkovich-z-21/1-2-08/7-2.webp?ts=1734380238" ] ] ] #original: array:6 [ "id" => 273020 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1158} "task" => array:2 [ "refs" => "227164" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "7-1.png" "alt" => null "width" => "705" "height" => 1680 "path" => "/media/algebra_08/mordkovich-z-21/1-2-08/7-1.webp?ts=1734380238" ] 1 => array:5 [ "name" => "7-2.png" "alt" => null "width" => "705" "height" => 1977 "path" => "/media/algebra_08/mordkovich-z-21/1-2-08/7-2.webp?ts=1734380238" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1102 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 264941 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1093 #items: array:1 [ 0 => App\Models\Edition {#1055 #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" => 725 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1053 …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" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1049 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_content_source" => null ] #original: array:21 [ "id" => 725 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1053 …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" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1049 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1054 …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" => "227164" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "7-1.jpg" "alt" => null "width" => "700" "height" => 357 "path" => "/media/algebra_08/mordkovich-z-21/2-2-08/7-1.webp?ts=1734377967" ] ] ] #original: array:6 [ "id" => 264941 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1093} "task" => array:2 [ "refs" => "227164" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "7-1.jpg" "alt" => null "width" => "700" "height" => 357 "path" => "/media/algebra_08/mordkovich-z-21/2-2-08/7-1.webp?ts=1734377967" ] ] ] #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\Element {#1101 #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" => 266509 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1122 #items: array:1 [ 0 => App\Models\Edition {#1116 #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" => 727 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 4" "field_order" => "5" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1141 …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" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "4-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1149 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1139 …2} "field_content_source" => null ] #original: array:21 [ "id" => 727 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 4" "field_order" => "5" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1141 …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" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "4-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1149 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1139 …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" => "227164" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "7-1.png" "alt" => null "width" => "715" "height" => 541 "path" => "/media/algebra_08/mordkovich-z-21/4-2-08/7-1.webp?ts=1734378443" ] ] ] #original: array:6 [ "id" => 266509 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1122} "task" => array:2 [ "refs" => "227164" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "7-1.png" "alt" => null "width" => "715" "height" => 541 "path" => "/media/algebra_08/mordkovich-z-21/4-2-08/7-1.webp?ts=1734378443" ] ] ] #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\Element {#1100 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 267074 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1140 #items: array:1 [ 0 => App\Models\Edition {#1150 #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" => 728 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 5" "field_order" => "6" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1155 …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" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "5-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1159 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1152 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1156 …2} "field_content_source" => null ] #original: array:21 [ "id" => 728 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 5" "field_order" => "6" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1155 …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" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "5-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1159 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1152 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1156 …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" => "227164" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "7-1.jpg" "alt" => null "width" => "2403" "height" => 1056 "path" => "/media/algebra_08/mordkovich-z-21/5-2-08/7-1.webp?ts=1734378338" ] ] ] #original: array:6 [ "id" => 267074 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1140} "task" => array:2 [ "refs" => "227164" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "7-1.jpg" "alt" => null "width" => "2403" "height" => 1056 "path" => "/media/algebra_08/mordkovich-z-21/5-2-08/7-1.webp?ts=1734378338" ] ] ] #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\Element {#1098 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1335481 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1165 #items: array:1 [ 0 => App\Models\Edition {#1166 #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" => 5110 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 6" "field_order" => "7" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1176 …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" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "6-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1177 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5110 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 6" "field_order" => "7" …16 ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "227164" "type" => "task" ] "text" => "<p><strong>а) 2;</strong></p><p>Чтобы представить заданные числа в виде степени с основанием 2, необходимо найти такой показатель степени $n$, что $2^n$ равно заданному числу. Для целых чисел, больших 1, показатель будет положительным. Для дробей вида $\frac{1}{x}$ используется свойство степени с отрицательным показателем: $a^{-n} = \frac{1}{a^n}$.</p><p>Представим каждое число из списка в виде степени числа 2:</p><p>$2 = 2^1$</p><p>$4 = 2 \times 2 = 2^2$</p><p>$8 = 2 \times 2 \times 2 = 2^3$</p><p>$16 = 2^4$</p><p>$32 = 2^5$</p><p>$64 = 2^6$</p><p>$128 = 2^7$</p><p>Теперь представим дроби:</p><p>$\frac{1}{2} = \frac{1}{2^1} = 2^{-1}$</p><p>$\frac{1}{4} = \frac{1}{2^2} = 2^{-2}$</p><p>$\frac{1}{8} = \frac{1}{2^3} = 2^{-3}$</p><p>$\frac{1}{32} = \frac{1}{2^5} = 2^{-5}$</p><p>$\frac{1}{128} = \frac{1}{2^7} = 2^{-7}$</p><p><b>Ответ:</b> $2 = 2^1$; $4 = 2^2$; $8 = 2^3$; $16 = 2^4$; $32 = 2^5$; $64 = 2^6$; $128 = 2^7$; $\frac{1}{2} = 2^{-1}$; $\frac{1}{4} = 2^{-2}$; $\frac{1}{8} = 2^{-3}$; $\frac{1}{32} = 2^{-5}$; $\frac{1}{128} = 2^{-7}$.</p><p><strong>б) $\frac{1}{2}$;</strong></p><p>Чтобы представить числа в виде степени с основанием $\frac{1}{2}$, воспользуемся свойством степени $(\frac{a}{b})^n = \frac{a^n}{b^n}$ и свойством степени с отрицательным показателем $(\frac{a}{b})^{-n} = (\frac{b}{a})^n$. Для нашего случая, $(\frac{1}{2})^{-n} = (\frac{2}{1})^n = 2^n$.</p><p>Представим целые числа из списка, используя $2^n = (\frac{1}{2})^{-n}$:</p><p>$2 = 2^1 = (\frac{1}{2})^{-1}$</p><p>$4 = 2^2 = (\frac{1}{2})^{-2}$</p><p>$8 = 2^3 = (\frac{1}{2})^{-3}$</p><p>$16 = 2^4 = (\frac{1}{2})^{-4}$</p><p>$32 = 2^5 = (\frac{1}{2})^{-5}$</p><p>$64 = 2^6 = (\frac{1}{2})^{-6}$</p><p>$128 = 2^7 = (\frac{1}{2})^{-7}$</p><p>Теперь представим дроби. Для них показатель степени будет положительным:</p><p>$\frac{1}{2} = (\frac{1}{2})^1$</p><p>$\frac{1}{4} = \frac{1^2}{2^2} = (\frac{1}{2})^2$</p><p>$\frac{1}{8} = \frac{1^3}{2^3} = (\frac{1}{2})^3$</p><p>$\frac{1}{32} = \frac{1^5}{2^5} = (\frac{1}{2})^5$</p><p>$\frac{1}{128} = \frac{1^7}{2^7} = (\frac{1}{2})^7$</p><p><b>Ответ:</b> $2 = (\frac{1}{2})^{-1}$; $4 = (\frac{1}{2})^{-2}$; $8 = (\frac{1}{2})^{-3}$; $16 = (\frac{1}{2})^{-4}$; $32 = (\frac{1}{2})^{-5}$; $64 = (\frac{1}{2})^{-6}$; $128 = (\frac{1}{2})^{-7}$; $\frac{1}{2} = (\frac{1}{2})^1$; $\frac{1}{4} = (\frac{1}{2})^2$; $\frac{1}{8} = (\frac{1}{2})^3$; $\frac{1}{32} = (\frac{1}{2})^5$; $\frac{1}{128} = (\frac{1}{2})^7$.</p>" ] #original: array:6 [ "id" => 1335481 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1165} "task" => array:2 [ "refs" => "227164" "type" => "task" ] "text" => "<p><strong>а) 2;</strong></p><p>Чтобы представить заданные числа в виде степени с основанием 2, необходимо найти такой показатель степени $n$, что $2^n$ равно заданному числу. Для целых чисел, больших 1, показатель будет положительным. Для дробей вида $\frac{1}{x}$ используется свойство степени с отрицательным показателем: $a^{-n} = \frac{1}{a^n}$.</p><p>Представим каждое число из списка в виде степени числа 2:</p><p>$2 = 2^1$</p><p>$4 = 2 \times 2 = 2^2$</p><p>$8 = 2 \times 2 \times 2 = 2^3$</p><p>$16 = 2^4$</p><p>$32 = 2^5$</p><p>$64 = 2^6$</p><p>$128 = 2^7$</p><p>Теперь представим дроби:</p><p>$\frac{1}{2} = \frac{1}{2^1} = 2^{-1}$</p><p>$\frac{1}{4} = \frac{1}{2^2} = 2^{-2}$</p><p>$\frac{1}{8} = \frac{1}{2^3} = 2^{-3}$</p><p>$\frac{1}{32} = \frac{1}{2^5} = 2^{-5}$</p><p>$\frac{1}{128} = \frac{1}{2^7} = 2^{-7}$</p><p><b>Ответ:</b> $2 = 2^1$; $4 = 2^2$; $8 = 2^3$; $16 = 2^4$; $32 = 2^5$; $64 = 2^6$; $128 = 2^7$; $\frac{1}{2} = 2^{-1}$; $\frac{1}{4} = 2^{-2}$; $\frac{1}{8} = 2^{-3}$; $\frac{1}{32} = 2^{-5}$; $\frac{1}{128} = 2^{-7}$.</p><p><strong>б) $\frac{1}{2}$;</strong></p><p>Чтобы представить числа в виде степени с основанием $\frac{1}{2}$, воспользуемся свойством степени $(\frac{a}{b})^n = \frac{a^n}{b^n}$ и свойством степени с отрицательным показателем $(\frac{a}{b})^{-n} = (\frac{b}{a})^n$. Для нашего случая, $(\frac{1}{2})^{-n} = (\frac{2}{1})^n = 2^n$.</p><p>Представим целые числа из списка, используя $2^n = (\frac{1}{2})^{-n}$:</p><p>$2 = 2^1 = (\frac{1}{2})^{-1}$</p><p>$4 = 2^2 = (\frac{1}{2})^{-2}$</p><p>$8 = 2^3 = (\frac{1}{2})^{-3}$</p><p>$16 = 2^4 = (\frac{1}{2})^{-4}$</p><p>$32 = 2^5 = (\frac{1}{2})^{-5}$</p><p>$64 = 2^6 = (\frac{1}{2})^{-6}$</p><p>$128 = 2^7 = (\frac{1}{2})^{-7}$</p><p>Теперь представим дроби. Для них показатель степени будет положительным:</p><p>$\frac{1}{2} = (\frac{1}{2})^1$</p><p>$\frac{1}{4} = \frac{1^2}{2^2} = (\frac{1}{2})^2$</p><p>$\frac{1}{8} = \frac{1^3}{2^3} = (\frac{1}{2})^3$</p><p>$\frac{1}{32} = \frac{1^5}{2^5} = (\frac{1}{2})^5$</p><p>$\frac{1}{128} = \frac{1^7}{2^7} = (\frac{1}{2})^7$</p><p><b>Ответ:</b> $2 = (\frac{1}{2})^{-1}$; $4 = (\frac{1}{2})^{-2}$; $8 = (\frac{1}{2})^{-3}$; $16 = (\frac{1}{2})^{-4}$; $32 = (\frac{1}{2})^{-5}$; $64 = (\frac{1}{2})^{-6}$; $128 = (\frac{1}{2})^{-7}$; $\frac{1}{2} = (\frac{1}{2})^1$; $\frac{1}{4} = (\frac{1}{2})^2$; $\frac{1}{8} = (\frac{1}{2})^3$; $\frac{1}{32} = (\frac{1}{2})^5$; $\frac{1}{128} = (\frac{1}{2})^7$.</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" => "227165" "type" => "task" ] "previous" => array:2 [ "refs" => "227163" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1104 #items: array:1 [ 0 => App\Models\Book {#1057} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1107 #items: array:1 [ 0 => App\Models\BookPage {#1109 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 865255 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "52" "field_url" => "/8-klass/algebra/mordkovich/page-2-52" "field_display_title" => "52" "field_folder" => "2" "field_image_name" => "52" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1111 #items: array:1 [ 0 => App\Models\Branch {#1124} ] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1042 #items: array:1 [ 0 => App\Models\Book {#1057} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1110 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1178 #items: array:1 [ 0 => App\Models\Element {#1175 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "865256" "type" => "book_page" ] "previous" => array:2 [ "refs" => "865254" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1285 #items: array:9 [ 0 => App\Models\Task {#1301 #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] } 1 => App\Models\Task {#1308 #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] } 2 => App\Models\Task {#1311 #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] } 3 => App\Models\Task {#1310 #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] } 4 => App\Models\Task {#1312 #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] } 5 => App\Models\Task {#1332 #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] } 6 => App\Models\Task {#1352 #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 {#1372 #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 {#1435 #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" => 865255 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "52" "field_url" => "/8-klass/algebra/mordkovich/page-2-52" "field_display_title" => "52" "field_folder" => "2" "field_image_name" => "52" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1111} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1042} "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 {#1110} "content" => Illuminate\Database\Eloquent\Collection {#1178} "next" => array:2 [ "refs" => "865256" "type" => "book_page" ] "previous" => array:2 [ "refs" => "865254" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1285} ] #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: 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№8.7 (с. 52)
Условие. №8.7 (с. 52)
скриншот условия
8.7 Представьте числа 2, 4, 8, 16, 32, 64, 128, $\frac{1}{2}$, $\frac{1}{4}$, $\frac{1}{8}$, $\frac{1}{32}$, $\frac{1}{128}$ в виде степени числа:
а) 2;
б) $\frac{1}{2}$.
Решение 1. №8.7 (с. 52)
Решение 2. №8.7 (с. 52)
Решение 4. №8.7 (с. 52)
Решение 5. №8.7 (с. 52)
Решение 6. №8.7 (с. 52)
а) 2;
Чтобы представить заданные числа в виде степени с основанием 2, необходимо найти такой показатель степени $n$, что $2^n$ равно заданному числу. Для целых чисел, больших 1, показатель будет положительным. Для дробей вида $\frac{1}{x}$ используется свойство степени с отрицательным показателем: $a^{-n} = \frac{1}{a^n}$.
Представим каждое число из списка в виде степени числа 2:
$2 = 2^1$
$4 = 2 \times 2 = 2^2$
$8 = 2 \times 2 \times 2 = 2^3$
$16 = 2^4$
$32 = 2^5$
$64 = 2^6$
$128 = 2^7$
Теперь представим дроби:
$\frac{1}{2} = \frac{1}{2^1} = 2^{-1}$
$\frac{1}{4} = \frac{1}{2^2} = 2^{-2}$
$\frac{1}{8} = \frac{1}{2^3} = 2^{-3}$
$\frac{1}{32} = \frac{1}{2^5} = 2^{-5}$
$\frac{1}{128} = \frac{1}{2^7} = 2^{-7}$
Ответ: $2 = 2^1$; $4 = 2^2$; $8 = 2^3$; $16 = 2^4$; $32 = 2^5$; $64 = 2^6$; $128 = 2^7$; $\frac{1}{2} = 2^{-1}$; $\frac{1}{4} = 2^{-2}$; $\frac{1}{8} = 2^{-3}$; $\frac{1}{32} = 2^{-5}$; $\frac{1}{128} = 2^{-7}$.
б) $\frac{1}{2}$;
Чтобы представить числа в виде степени с основанием $\frac{1}{2}$, воспользуемся свойством степени $(\frac{a}{b})^n = \frac{a^n}{b^n}$ и свойством степени с отрицательным показателем $(\frac{a}{b})^{-n} = (\frac{b}{a})^n$. Для нашего случая, $(\frac{1}{2})^{-n} = (\frac{2}{1})^n = 2^n$.
Представим целые числа из списка, используя $2^n = (\frac{1}{2})^{-n}$:
$2 = 2^1 = (\frac{1}{2})^{-1}$
$4 = 2^2 = (\frac{1}{2})^{-2}$
$8 = 2^3 = (\frac{1}{2})^{-3}$
$16 = 2^4 = (\frac{1}{2})^{-4}$
$32 = 2^5 = (\frac{1}{2})^{-5}$
$64 = 2^6 = (\frac{1}{2})^{-6}$
$128 = 2^7 = (\frac{1}{2})^{-7}$
Теперь представим дроби. Для них показатель степени будет положительным:
$\frac{1}{2} = (\frac{1}{2})^1$
$\frac{1}{4} = \frac{1^2}{2^2} = (\frac{1}{2})^2$
$\frac{1}{8} = \frac{1^3}{2^3} = (\frac{1}{2})^3$
$\frac{1}{32} = \frac{1^5}{2^5} = (\frac{1}{2})^5$
$\frac{1}{128} = \frac{1^7}{2^7} = (\frac{1}{2})^7$
Ответ: $2 = (\frac{1}{2})^{-1}$; $4 = (\frac{1}{2})^{-2}$; $8 = (\frac{1}{2})^{-3}$; $16 = (\frac{1}{2})^{-4}$; $32 = (\frac{1}{2})^{-5}$; $64 = (\frac{1}{2})^{-6}$; $128 = (\frac{1}{2})^{-7}$; $\frac{1}{2} = (\frac{1}{2})^1$; $\frac{1}{4} = (\frac{1}{2})^2$; $\frac{1}{8} = (\frac{1}{2})^3$; $\frac{1}{32} = (\frac{1}{2})^5$; $\frac{1}{128} = (\frac{1}{2})^7$.
Другие задания:
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