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Краткие ответы

1. \(2\)
2. \(81\)
3. \(6\)
4. \(2\)
5. \(0{,}7\)
6. \(1\)
7. \(216\)
8. \(28561\)
9. \(\dfrac{1}{25}\)

В № 9 последний показатель читается как \(-3(\sqrt7+1)\).

Решение

1. Представим корни как степени:

\[ 2\cdot\frac{\sqrt[4]{a}\cdot\sqrt[12]{a}}{\sqrt[3]{a}} = 2a^{\frac14+\frac1{12}-\frac13} = 2a^{\frac3{12}+\frac1{12}-\frac4{12}} = 2a^0=2. \]

2.

\[ 3^{0{,}34}\cdot27^{1{,}22} = 3^{0{,}34}\cdot(3^3)^{1{,}22} = 3^{0{,}34+3{,}66} = 3^4=81. \]

3.

\[ \frac{\sqrt{4{,}8}\cdot\sqrt{4{,}2}}{\sqrt{0{,}56}} = \sqrt{\frac{4{,}8\cdot4{,}2}{0{,}56}} = \sqrt{\frac{20{,}16}{0{,}56}} = \sqrt{36}=6. \]

4.

\[ \frac{\sqrt[5]{14}\cdot\sqrt[5]{16}}{\sqrt[5]{7}} = \sqrt[5]{\frac{14\cdot16}{7}} = \sqrt[5]{32}=2, \]

так как \(2^5=32\).

5.

\[ \frac{\sqrt{2{,}1}\cdot\sqrt{1{,}4}}{\sqrt6} = \sqrt{\frac{2{,}1\cdot1{,}4}{6}} = \sqrt{0{,}49}=0{,}7. \]

Проверка: \(0{,}7^2=0{,}49\).

6.

\[ \frac{3(3x^3)^2(5y)^3}{(15x^2y)^3} = \frac{3\cdot9x^6\cdot125y^3}{3375x^6y^3} = \frac{3375x^6y^3}{3375x^6y^3} =1. \]

7.

\[ \frac{6^{3\sqrt2+2}\cdot6^{2\sqrt2}}{6^{5\sqrt2-1}} = 6^{(3\sqrt2+2)+2\sqrt2-(5\sqrt2-1)} = 6^3=216. \]

8.

\[ (13x^2)^5:(13x^{10}) = \frac{13^5x^{10}}{13x^{10}} = 13^4=28561. \]

9.

\[ 5^{4\sqrt7-1}\cdot5^{2-\sqrt7}\cdot5^{-3(\sqrt7+1)} \]

\[ = 5^{\,4\sqrt7-1+2-\sqrt7-3\sqrt7-3} = 5^{-2} = \frac1{25}. \]