123. Выполните действия:
a) √3 (√27 +0,4√/75);
6) (√20-6√5 +√45) √5;
г) (√/15 - √6)2 + (√/3 + /30)2.
B) (2√8 + 3√5-7√2)(2√2+2√5);
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Ответы
Ответ:
Let's simplify each of these expressions step by step:
a) √3 (√27 + 0.4√75):
First, let's simplify the square roots:
√27 = √(9 * 3) = 3√3
√75 = √(25 * 3) = 5√3
Now, we can rewrite the expression:
√3 (3√3 + 0.4 * 5√3)
Next, distribute the √3 into the parentheses:
3√3 * √3 + 0.4 * 5√3 * √3
Now, multiply the terms inside the square roots:
3 * 3 + 0.4 * 5 * 3
Simplify further:
9 + 6
Combine like terms:
15
So, the simplified expression is 15.
(√20 - 6√5 + √45)√5:
Let's simplify the square roots first:
√20 = √(4 * 5) = 2√5
√45 = √(9 * 5) = 3√5
Now, rewrite the expression:
(2√5 - 6√5 + 3√5)√5
Now, combine like terms inside the parentheses:
(2√5 - 6√5 + 3√5)√5 = (-1√5)√5
Multiplying the square roots:
-1 * 5 = -5
So, the simplified expression is -5.
г) (√15 - √6)² + (√3 + √30)²:
Let's simplify each square separately:
(√15 - √6)²:
Expand this using the formula (a - b)² = a² - 2ab + b²:
(√15)² - 2(√15)(√6) + (√6)²
15 - 2√(15 * 6) + 6
(√3 + √30)²:
Expand using the same formula:
(√3)² + 2(√3)(√30) + (√30)²
3 + 2√(3 * 30) + 30
Now, add the two simplified squares together:
(15 - 2√90 + 6) + (3 + 2√90 + 30)
Combine like terms:
(15 + 6 + 3) + (-2√90 + 2√90) + 30
Simplify further:
24 + 0 + 30
So, the simplified expression is 54.
B) (2√8 + 3√5 - 7√2)(2√2 + 2√5):
First, simplify the square roots:
√8 = √(4 * 2) = 2√2
√5 = √5
Now, rewrite the expression:
(2√2 + 3√5 - 7√2)(2√2 + 2√5)
Now, distribute and multiply the terms:
(2√2 * 2√2) + (2√2 * 2√5) + (3√5 * 2√2) + (3√5 * 2√5) + (-7√2 * 2√2) + (-7√2 * 2√5)
Simplify each term:
4 * 2 + 4√10 + 6√10 + 6 * 5 - 14 * 2√2 - 14 * 2√5
Now, combine like terms:
8 + 10√10 + 30 - 28√2 - 28√5
Combine the constants:
(8 + 30) + (-28√2 - 28√5) + 10√10
38 - 28(√2 + √5) + 10√10
So, the simplified expression is 38 - 28(√2 + √5) + 10√10.
Пошаговое объяснение: