Автор - refns

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Ответ

Автор - MaxikMK

- Задание №1:

1) left(sqrt[6]{7^3}right)^2 = left(7^frac{3}{6}right)^2 = 7^{frac{1}{2}cdot2} = 7^1 = 7.

2) left(sqrt[6]{9}right)^{-3} = left(sqrt[6]{3^2}right)^{-3} = left(3^frac{2}{6}right)^{-3} = 3^{frac{1}{3}cdot(-3)} = 3^{-1} = dfrac{1}{3}.

3) left(sqrt[10]{32}right)^{2} = left(sqrt[10]{2^5}right)^{2} = left(2^frac{5}{10}right)^{2} = 2^{frac{1}{2}cdot2} = 2^1 = 2.

4) left(sqrt[8]{16}right)^{-4} = left(sqrt[8]{2^4}right)^{-4} = left(2^frac{4}{8}right)^{-4} = 2^{frac{1}{2}cdot(-4)} = 2^{-2} = dfrac{1}{4}.

- Задание №2:

1) sqrt{sqrt[3]{729}} = sqrt[6]{27^2} = sqrt[6]{(3^3)^2} = sqrt[6]{3^{3*2}} = sqrt[6]{3^6} = 3.

2) sqrt{sqrt{1024}} = sqrt[4]{2^{10}} = 2^frac{10}{4} = 2^frac{5}{2} = sqrt{2^5} = sqrt{16 * 2} = 4sqrt{2}.

3) sqrt[3]{sqrt[3]{9}}cdotsqrt[9]{3^7} = sqrt[9]{3^2}cdotsqrt[9]{3^7} = 3^frac{2}{9} * 3^frac{7}{9} = 3^{frac{2}{9} + frac{7}{9}} = 3^1 = 3.

4) sqrt[4]{sqrt[3]{25}}cdotsqrt[6]{5} = sqrt[12]{5^2}cdotsqrt[6]{5} = 5^frac{2}{12}cdot5^frac{1}{6} = 5^frac{1}{6}cdot5^frac{1}{6} = 5^{frac{1}{6} + frac{1}{6}} = 5^frac{2}{6} = 5^frac{1}{3} = sqrt[3]{5}.

- Задание №3:

1) left(sqrt[3]{x}right)^6 = x^frac{6}{3} = x^2.

2) left(sqrt[3]{y^2}right)^3 = left(y^2right)^frac{3}{3} = left(y^2right)^1 = y^2.

3) left(sqrt{a}cdotsqrt[3]{b}right)^6 = left(sqrt{a}right)^6cdotleft(sqrt[3]{b}right)^6 = a^frac{6}{2}cdot b^frac{6}{3} = a^3b^2.

4) left(sqrt[3]{a^2}cdotsqrt[4]{b^3}right)^{12} = left(a^frac{2}{3}cdot b^frac{3}{4}right)^{12} = left(a^frac{2}{3}right)^{12}cdotleft(b^frac{3}{4}right)^{12} = a^{frac{2}{3}cdot12}cdot b^{frac{3}{4}cdot12} = a^8b^9.

5) left(sqrt{sqrt[3]{a^2b}}right)^6 = left(sqrt[6]{a^2b}right)^6 = left(a^2bright)^frac{6}{6} = left(a^2bright)^1 = a^2b.

6) left(sqrt[3]{sqrt[4]{27a^3}}right)^4 = left(sqrt[12]{left(3aright)^3}right)^4 = left(sqrt[12]{3a}right)^{12} = left(3aright)^frac{12}{12} = left(3aright)^1 = 3a.

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