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Квадратные корни

Презентация содержит определение и свойства арифметического квадратного корня. Рассматриваются примеры.
17.10.2020

Содержимое разработки

Урок по теме «Квадратные корни»  Алгебра, 8 класс    МОУ СОШ № 31, г.Ростов-на-Дону, учитель математики Кряквина Л.Н. 2005 год

Урок по теме «Квадратные корни» Алгебра, 8 класс

МОУ СОШ № 31,

г.Ростов-на-Дону,

учитель математики

Кряквина Л.Н.

2005 год

Содержание   1. Понятие квадратного корня из неотрицательного числа. 2. Функция y=√x , ее свойства и график. 3. Свойства квадратных корней. 4. Математический диктант. 5. Вопросы для самоконтроля.

Содержание

  • 1. Понятие квадратного корня из неотрицательного числа.
  • 2. Функция y=√x , ее свойства и график.
  • 3. Свойства квадратных корней.
  • 4. Математический диктант.
  • 5. Вопросы для самоконтроля.
Понятие квадратного корня из неотрицательного числа Квадратным корнем из неотрицательного числа a называют такое неотрицательное число, квадрат которого равен a . Это число обозначают √ a , число при этом называют подкоренным числом.

Понятие квадратного корня из неотрицательного числа

  • Квадратным корнем из неотрицательного числа a называют такое неотрицательное число, квадрат которого равен a . Это число обозначают √ a , число при этом называют подкоренным числом.
Определения Операцию нахождения квадратного корня из неотрицательного числа называют извлечением квадратного корня. Эта операция является обратной по отношению к возведению в квадрат. Например, 5 2 =25 и √25=5.

Определения

  • Операцию нахождения квадратного корня из неотрицательного числа называют извлечением квадратного корня. Эта операция является обратной по отношению к возведению в квадрат. Например, 5 2 =25 и √25=5.
Используем следующую таблицу для построения графика функции X 0 1 Y 0 1 4 6,25 2 2,5 9 16 3 4 Построим найденные точки (0;0), (1;1), (4;2), (6,25;2,5) , (9;3) , (16;4)  в координатной плоскости. Они располагаются на некоторой линии, начертим ее. Получили график функции y= √x.

Используем следующую таблицу для построения графика функции

X 0

1

Y 0

1

4

6,25

2

2,5

9

16

3

4

Построим найденные точки (0;0), (1;1), (4;2), (6,25;2,5) , (9;3) , (16;4)

в координатной плоскости. Они располагаются на некоторой

линии, начертим ее. Получили график функции y= √x.

График функции квадратного корня Λ ›

График функции квадратного корня

Λ

0 при х 0 . 3. Функция непрерывна на луче [ 0 ;+∞) . 4. Функция ограничена снизу и не ограничена сверху. 5. у наим =0 (достигается при х=0), у наиб не существует. " width="640"

Функция квадратного корня

  • Свойства функции y= √ x :
  • 1. Область определения функции – луч [ 0 ;+∞) .
  • 2. у=0 при х=0 ; у 0 при х 0 .
  • 3. Функция непрерывна на луче [ 0 ;+∞) .
  • 4. Функция ограничена снизу и не ограничена сверху.
  • 5. у наим =0 (достигается при х=0), у наиб не существует.
Свойства квадратных корней

Свойства квадратных корней

Математический диктант Учитель предлагает учащимся плакат с утверждениями, некоторые из которых содержат ошибки или неточности. Учащиеся должны определить верные и неверные утверждения. Свой ответ они должны аргументировать, ссылаясь на известные свойства и правила.

Математический диктант

  • Учитель предлагает учащимся плакат с утверждениями, некоторые из которых содержат ошибки или неточности. Учащиеся должны
  • определить верные и неверные утверждения. Свой ответ они должны аргументировать, ссылаясь на известные свойства и правила.

Текст математического диктанта

?

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  • Если m
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  • m 4 = size 12{ sqrt {m rSup { size 8{4} } } ={}} {} VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAATMFAAA/ AgAALQAAAJYAAQACAAAACQCLAAEAAgAAAP//gQABABAAAAAAAAAAAAAAADIFAAA+AgAAlQAB AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEAPAAAAAMANgAAAAoAT3BlblN5bWJv bAAApgEAAFwCAAD//wAAAAAFAAAAAAAAAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCH AAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABoAAABqAAAA9wEAAAEAAAAaIuEAAAAAAAEA AQAaIowAAQAAAAAAiwABAAIAAAAfAIUAAQAFAAAAAAAAAAGEAAEABQAAAAAAAAAAigABADwA AAADADYAAAAKAE9wZW5TeW1ib2wAAKYBAABcAgAA//8AAAAABQAAAAAAAAD/AwAAAAAAAAAA AP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAGcAAQAQAAAAWAEA ABoAAABHAwAAKAAAAIwAAQAAAAAAiwABAAIAAAAfAIoAAQBCAAAAAwA8AAAAEABMaWJlcmF0 aW9uIFNlcmlmAAAAAAAApgEAAAAAAwAAAAUAAAAAAAIA/wMAAAAAAAAAAAD/AwAAAAAAiAAB AAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAHIBAADcAQAAAQBtMQEA AAAAAQABAG0AjAABAAAAAACLAAEAAgAAAB8AigABAEIAAAADADwAAAAQAExpYmVyYXRpb24g U2VyaWYAAAAAAAAZAQAAAAADAAAABQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAA AAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAXAAAAsAIAACMBAAABADSNAAAAAAAB AAEANACMAAEAAAAAAIsAAQACAAAAHwCKAAEAPAAAAAMANgAAAAoAT3BlblN5bWJvbAAAAAAA AKYBAAD//wAAAAAFAAAAAAAAAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAA AP////8AhgABAAQAAAAAAAAAcgACABoAAABPAwAA3AEAAAEAAAA9AE8BAAAAAAEAAQA9AIwA AQAAAAAAlQABAAQAAAAAAAAAlgABAAIAAAAJAIwAAQAAAAAA m 2 .
  • a a 2 size 12{ sqrt {a rSup { size 8{2} } } } {} VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAWYDAAA/ AgAAJgAAAJYAAQACAAAACQCLAAEAAgAAAP//gQABABAAAAAAAAAAAAAAAGUDAAA+AgAAlQAB AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEAPAAAAAMANgAAAAoAT3BlblN5bWJv bAAApgEAAFwCAAD//wAAAAAFAAAAAAAAAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCH AAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABoAAABqAAAA9wEAAAEAAAAaIuEAAAAAAAEA AQAaIowAAQAAAAAAiwABAAIAAAAfAIUAAQAFAAAAAAAAAAGEAAEABQAAAAAAAAAAigABADwA AAADADYAAAAKAE9wZW5TeW1ib2wAAKYBAABcAgAA//8AAAAABQAAAAAAAAD/AwAAAAAAAAAA AP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAGcAAQAQAAAAWAEA ABoAAADdAgAAKAAAAIwAAQAAAAAAiwABAAIAAAAfAIoAAQBCAAAAAwA8AAAAEABMaWJlcmF0 aW9uIFNlcmlmAAAAAAAApgEAAAAAAwAAAAUAAAAAAAIA/wMAAAAAAAAAAAD/AwAAAAAAiAAB AAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAHIBAADcAQAAAQBh0wAA AAAAAQABAGEAjAABAAAAAACLAAEAAgAAAB8AigABAEIAAAADADwAAAAQAExpYmVyYXRpb24g U2VyaWYAAAAAAAAZAQAAAAADAAAABQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAA AAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAXAAAARgIAACMBAAABADKNAAAAAAAB AAEAMgCMAAEAAAAAAJUAAQAEAAAAAAAAAJYAAQACAAAACQCMAAEAAAAAAA== = a 2 .
Тест

Тест

Задание № 1 Чему равен √3364? 1. 58 2. 52 3. 56 4. 62

Задание № 1

  • Чему равен √3364?
  • 1. 58
  • 2. 52
  • 3. 56
  • 4. 62
Задание № 2 Чему равно значение числового выражения ⅔ ٠ √ 150 ٠ 0,25 ٠ √ 6 ? 1. 5 2 30 3. 12 4. 6

Задание № 2

  • Чему равно значение числового выражения ⅔ ٠ √ 150 ٠ 0,25 ٠ √ 6 ?
  • 1. 5
  • 2 30
  • 3. 12
  • 4. 6
Задание № 3 Упростите выражение √75 b 7  : √3 b 3 . 1. 25b 4 2. 3b 4 3. -3b 4 4. 5b 2

Задание № 3

  • Упростите выражение √75 b 7 : √3 b 3 .
  • 1. 25b 4
  • 2. 3b 4
  • 3. -3b 4
  • 4. 5b 2
Задание № 4 Найдите наименьшее значение функции y= √x+1 на отрезке [-1,1]. 1. 2 2. -2 3. 0 4. 1

Задание № 4

  • Найдите наименьшее значение функции y= √x+1 на отрезке [-1,1].
  • 1. 2
  • 2. -2
  • 3. 0
  • 4. 1

Задания для самоконтроля

  • Вычислите: а) 2 6 7 2 size 12{ sqrt {2 rSup { size 8{6} } 7 rSup { size 8{2} } } } {} VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAfUEAAA/ AgAANAAAAJYAAQACAAAACQCLAAEAAgAAAP//gQABABAAAAAAAAAAAAAAAPQEAAA+AgAAlQAB AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEAPAAAAAMANgAAAAoAT3BlblN5bWJv bAAApgEAAFwCAAD//wAAAAAFAAAAAAAAAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCH AAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABoAAABqAAAA9wEAAAEAAAAaIuEAAAAAAAEA AQAaIowAAQAAAAAAiwABAAIAAAAfAIUAAQAFAAAAAAAAAAGEAAEABQAAAAAAAAAAigABADwA AAADADYAAAAKAE9wZW5TeW1ib2wAAKYBAABcAgAA//8AAAAABQAAAAAAAAD/AwAAAAAAAAAA AP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAGcAAQAQAAAAWAEA ABoAAABsBAAAKAAAAIwAAQAAAAAAiwABAAIAAAAfAIoAAQBCAAAAAwA8AAAAEABMaWJlcmF0 aW9uIFNlcmlmAAAAAAAApgEAAAAAAwAAAAUAAAAAAAAA/wMAAAAAAAAAAAD/AwAAAAAAiAAB AAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAHIBAADcAQAAAQAy0wAA AAAAAQABADIAjAABAAAAAACLAAEAAgAAAB8AigABAEIAAAADADwAAAAQAExpYmVyYXRpb24g U2VyaWYAAAAAAAAZAQAAAAADAAAABQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAA AAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAXAAAARgIAACMBAAABADaNAAAAAAAB AAEANgCMAAEAAAAAAIsAAQACAAAAHwCKAAEAQgAAAAMAPAAAABAATGliZXJhdGlvbiBTZXJp ZgAAAAAAAKYBAAAAAAMAAAAFAAAAAAAAAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCH AAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABcAAAD/AgAA3AEAAAEAN9MAAAAAAAEAAQA3 AIwAAQAAAAAAiwABAAIAAAAfAIoAAQBCAAAAAwA8AAAAEABMaWJlcmF0aW9uIFNlcmlmAAAA AAAAGQEAAAAAAwAAAAUAAAAAAAAA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcAAQAF AAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAANMDAAAjAQAAAQAyjQAAAAAAAQABADIAjAAB AAAAAACVAAEABAAAAAAAAACWAAEAAgAAAAkAjAABAAAAAAA= , б) 13 4 9 . size 12{ sqrt {"13" { {4} over {9} } "." } } {} VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAaMFAAA9 BAAAPQAAAJYAAQACAAAACQCLAAEAAgAAAP//gQABABAAAAAAAAAAAAAAAKIFAAA8BAAAlQAB AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEAPAAAAAMANgAAAAoAT3BlblN5bWJv bAAApgEAANYEAAD//wAAAAAFAAAAAAAAAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCH AAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABoAAABqAAAA7QMAAAEAAAAaIuEAAAAAAAEA AQAaIowAAQAAAAAAiwABAAIAAAAfAIUAAQAFAAAAAAAAAAGEAAEABQAAAAAAAAAAigABADwA AAADADYAAAAKAE9wZW5TeW1ib2wAAKYBAADWBAAA//8AAAAABQAAAAAAAAD/AwAAAAAAAAAA AP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAGcAAQAQAAAAWAEA ABoAAAAaBQAAKAAAAIwAAQAAAAAAiwABAAIAAAAfAIoAAQBCAAAAAwA8AAAAEABMaWJlcmF0 aW9uIFNlcmlmAAAAAAAApgEAAAAAAwAAAAUAAAAAAAAA/wMAAAAAAAAAAAD/AwAAAAAAiAAB AAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAGgAAAHIBAADKAgAAAgAxM6YB AAAAAAIAAgAxADMAjAABAAAAAACLAAEAAgAAAB8AigABAEIAAAADADwAAAAQAExpYmVyYXRp b24gU2VyaWYAAAAAAACmAQAAAAADAAAABQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAAAACIAAEA AgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAXAAAAaQMAANwBAAABADTTAAAA AAABAAEANACMAAEAAAAAAIsAAQACAAAAHwCFAAEABQAAAAAAAAABhAABAAUAAAAAAAAAAIoA AQA8AAAAAwA2AAAACgBPcGVuU3ltYm9sAAAAAAAApgEAAP//AAAAAAUAAAAAAAAA/wMAAAAA AAAAAAD/AwAAAAAAiAABAAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAABnAAEAEAAA AE8DAABGAgAAaAQAAFoCAACMAAEAAAAAAIsAAQACAAAAHwCKAAEAQgAAAAMAPAAAABAATGli ZXJhdGlvbiBTZXJpZgAAAAAAAKYBAAAAAAMAAAAFAAAAAAAAAP8DAAAAAAAAAAAA/wMAAAAA AIgAAQACAAAAAQCHAAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABcAAABpAwAA7QMAAAEA OdMAAAAAAAEAAQA5AIwAAQAAAAAAiwABAAIAAAAfAIoAAQBCAAAAAwA8AAAAEABMaWJlcmF0 aW9uIFNlcmlmAAAAAAAApgEAAAAAAwAAAAUAAAAAAAAA/wMAAAAAAAAAAAD/AwAAAAAAiAAB AAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAKcEAADKAgAAAQAuagAA AAAAAQABAC4AjAABAAAAAACVAAEABAAAAAAAAACWAAEAAgAAAAkAjAABAAAAAAA=
  • Сравните: 5 3 size 12{ sqrt {3} } {} VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAc8CAAD5 AQAAHwAAAJYAAQACAAAACQCLAAEAAgAAAP//gQABABAAAAAAAAAAAAAAAM4CAAD4AQAAlQAB AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEAPAAAAAMANgAAAAoAT3BlblN5bWJv bAAApgEAAAUCAAD//wAAAAAFAAAAAAAAAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCH AAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABoAAABqAAAAwgEAAAEAAAAaIuEAAAAAAAEA AQAaIowAAQAAAAAAiwABAAIAAAAfAIUAAQAFAAAAAAAAAAGEAAEABQAAAAAAAAAAigABADwA AAADADYAAAAKAE9wZW5TeW1ib2wAAKYBAAAFAgAA//8AAAAABQAAAAAAAAD/AwAAAAAAAAAA AP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAGcAAQAQAAAAWAEA ABoAAABGAgAAKAAAAIwAAQAAAAAAiwABAAIAAAAfAIoAAQBCAAAAAwA8AAAAEABMaWJlcmF0 aW9uIFNlcmlmAAAAAAAApgEAAAAAAwAAAAUAAAAAAAAA/wMAAAAAAAAAAAD/AwAAAAAAiAAB AAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAHIBAACnAQAAAQAz0wAA AAAAAQABADMAjAABAAAAAACVAAEABAAAAAAAAACWAAEAAgAAAAkAjAABAAAAAAA= и 6 2 size 12{ sqrt {2} } {} VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAcoCAAD5 AQAAHwAAAJYAAQACAAAACQCLAAEAAgAAAP//gQABABAAAAAAAAAAAAAAAMkCAAD4AQAAlQAB AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEAPAAAAAMANgAAAAoAT3BlblN5bWJv bAAApgEAAAUCAAD//wAAAAAFAAAAAAAAAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCH AAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABoAAABqAAAAwgEAAAEAAAAaIuEAAAAAAAEA AQAaIowAAQAAAAAAiwABAAIAAAAfAIUAAQAFAAAAAAAAAAGEAAEABQAAAAAAAAAAigABADwA AAADADYAAAAKAE9wZW5TeW1ib2wAAKYBAAAFAgAA//8AAAAABQAAAAAAAAD/AwAAAAAAAAAA AP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAGcAAQAQAAAAWAEA ABoAAABBAgAAKAAAAIwAAQAAAAAAiwABAAIAAAAfAIoAAQBCAAAAAwA8AAAAEABMaWJlcmF0 aW9uIFNlcmlmAAAAAAAApgEAAAAAAwAAAAUAAAAAAAAA/wMAAAAAAAAAAAD/AwAAAAAAiAAB AAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAHIBAACnAQAAAQAy0wAA AAAAAQABADIAjAABAAAAAACVAAEABAAAAAAAAACWAAEAAgAAAAkAjAABAAAAAAA= .

а)75 x 3 y 6 , size 12{ sqrt {x rSup { size 8{3} } y rSup { size 8{6} } ,} } {} VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAdMFAAA/ AgAAOwAAAJYAAQACAAAACQCLAAEAAgAAAP//gQABABAAAAAAAAAAAAAAANIFAAA+AgAAlQAB AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEAPAAAAAMANgAAAAoAT3BlblN5bWJv bAAApgEAAFwCAAD//wAAAAAFAAAAAAAAAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCH AAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABoAAABqAAAA9wEAAAEAAAAaIuEAAAAAAAEA AQAaIowAAQAAAAAAiwABAAIAAAAfAIUAAQAFAAAAAAAAAAGEAAEABQAAAAAAAAAAigABADwA AAADADYAAAAKAE9wZW5TeW1ib2wAAKYBAABcAgAA//8AAAAABQAAAAAAAAD/AwAAAAAAAAAA AP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAGcAAQAQAAAAWAEA ABoAAABKBQAAKAAAAIwAAQAAAAAAiwABAAIAAAAfAIoAAQBCAAAAAwA8AAAAEABMaWJlcmF0 aW9uIFNlcmlmAAAAAAAApgEAAAAAAwAAAAUAAAAAAAIA/wMAAAAAAAAAAAD/AwAAAAAAiAAB AAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAI0BAADcAQAAAQB4uwAA AAAAAQABAHgAjAABAAAAAACLAAEAAgAAAB8AigABAEIAAAADADwAAAAQAExpYmVyYXRpb24g U2VyaWYAAAAAAAAZAQAAAAADAAAABQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAA AAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAXAAAARgIAACMBAAABADONAAAAAAAB AAEAMwCMAAEAAAAAAIsAAQACAAAAHwCKAAEAQgAAAAMAPAAAABAATGliZXJhdGlvbiBTZXJp ZgAAAAAAAKYBAAAAAAMAAAAFAAAAAAACAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCH AAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABcAAABPAwAA3AEAAAEAebsAAAAAAAEAAQB5 AIwAAQAAAAAAiwABAAIAAAAfAIoAAQBCAAAAAwA8AAAAEABMaWJlcmF0aW9uIFNlcmlmAAAA AAAAGQEAAAAAAwAAAAUAAAAAAAAA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcAAQAF AAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAACIEAAAjAQAAAQA2jQAAAAAAAQABADYAjAAB AAAAAACLAAEAAgAAAB8AigABAEIAAAADADwAAAAQAExpYmVyYXRpb24gU2VyaWYAAAAAAACm AQAAAAADAAAABQAAAAAAAgD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/ ////AIYAAQAEAAAAAAAAAHIAAgAXAAAA3AQAANwBAAABACxqAAAAAAABAAEALACMAAEAAAAA AJUAAQAEAAAAAAAAAJYAAQACAAAACQCMAAEAAAAAAA== где y

Упростите выражение: 2 2 + 50 − 98 size 12{ sqrt {2} + sqrt {"50"} - sqrt {"98"} } {} VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAbEKAAD5 AQAAWwAAAJYAAQACAAAACQCLAAEAAgAAAP//gQABABAAAAAAAAAAAAAAALAKAAD4AQAAlQAB AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEAPAAAAAMANgAAAAoAT3BlblN5bWJv bAAApgEAAAUCAAD//wAAAAAFAAAAAAAAAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCH AAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABoAAABqAAAAwgEAAAEAAAAaIuEAAAAAAAEA AQAaIowAAQAAAAAAiwABAAIAAAAfAIUAAQAFAAAAAAAAAAGEAAEABQAAAAAAAAAAigABADwA AAADADYAAAAKAE9wZW5TeW1ib2wAAKYBAAAFAgAA//8AAAAABQAAAAAAAAD/AwAAAAAAAAAA AP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAGcAAQAQAAAAWAEA ABoAAABBAgAAKAAAAIwAAQAAAAAAiwABAAIAAAAfAIoAAQBCAAAAAwA8AAAAEABMaWJlcmF0 aW9uIFNlcmlmAAAAAAAApgEAAAAAAwAAAAUAAAAAAAAA/wMAAAAAAAAAAAD/AwAAAAAAiAAB AAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAHIBAACnAQAAAQAy0wAA AAAAAQABADIAjAABAAAAAACLAAEAAgAAAB8AigABADwAAAADADYAAAAKAE9wZW5TeW1ib2wA AAAAAACmAQAA//8AAAAABQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwAB AAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAaAAAARgIAAKcBAAABAAAAKwD2AAAAAAABAAEA KwCMAAEAAAAAAIsAAQACAAAAHwCKAAEAPAAAAAMANgAAAAoAT3BlblN5bWJvbAAApgEAAAUC AAD//wAAAAAFAAAAAAAAAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP// //8AhgABAAQAAAAAAAAAcgACABoAAABpAwAAwgEAAAEAAAAaIuEAAAAAAAEAAQAaIowAAQAA AAAAiwABAAIAAAAfAIUAAQAFAAAAAAAAAAGEAAEABQAAAAAAAAAAigABADwAAAADADYAAAAK AE9wZW5TeW1ib2wAAKYBAAAFAgAA//8AAAAABQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAAAACI AAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAGcAAQAQAAAAPQQAABoAAAD9BQAA KAAAAIwAAQAAAAAAiwABAAIAAAAfAIoAAQBCAAAAAwA8AAAAEABMaWJlcmF0aW9uIFNlcmlm AAAAAAAApgEAAAAAAwAAAAUAAAAAAAAA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcA AQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAGgAAAFcEAACnAQAAAgA1MKYBAAAAAAIAAgA1 ADAAjAABAAAAAACLAAEAAgAAAB8AigABADwAAAADADYAAAAKAE9wZW5TeW1ib2wAAAAAAACm AQAA//8AAAAABQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/ ////AIYAAQAEAAAAAAAAAHIAAgAaAAAAGQYAAKcBAAABAAAAEiJPAQAAAAABAAEAEiKMAAEA AAAAAIsAAQACAAAAHwCKAAEAPAAAAAMANgAAAAoAT3BlblN5bWJvbAAApgEAAAUCAAD//wAA AAAFAAAAAAAAAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgAB AAQAAAAAAAAAcgACABoAAABxBwAAwgEAAAEAAAAaIuEAAAAAAAEAAQAaIowAAQAAAAAAiwAB AAIAAAAfAIUAAQAFAAAAAAAAAAGEAAEABQAAAAAAAAAAigABADwAAAADADYAAAAKAE9wZW5T eW1ib2wAAKYBAAAFAgAA//8AAAAABQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAA AAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAGcAAQAQAAAAXwgAABoAAAAmCgAAKAAAAIwA AQAAAAAAiwABAAIAAAAfAIoAAQBCAAAAAwA8AAAAEABMaWJlcmF0aW9uIFNlcmlmAAAAAAAA pgEAAAAAAwAAAAUAAAAAAAAA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcAAQAFAAAA /////wCGAAEABAAAAAAAAAByAAIAGgAAAHoIAACnAQAAAgA5OKYBAAAAAAIAAgA5ADgAjAAB AAAAAACVAAEABAAAAAAAAACWAAEAAgAAAAkAjAABAAAAAAA= .

К сожалению, Вы ошиблись!

К сожалению, Вы ошиблись!

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Курсы повышения квалификации

Развитие пространственных представлений школьников в обучении математике в условиях реализации ФГОС

Продолжительность 36 часов
Документ: Удостоверение о повышении квалификации
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