Question

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Answer 1

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

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

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

- Задание №3:

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

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

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

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

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

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