Problems from Powers of fractional exponent

Development:

$4^{\frac{3}{7}}=\sqrt[7]{4^3}$

$8^{\frac{17}{5}}=\sqrt[5]{8^{17}}$

Solution:

$\sqrt[7]{4^3},\sqrt[5]{8^{17}}$

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Development:

$${\displaystyle \frac{\sqrt[4]{3^{23}\cdot 3\cdot (3^2{)}^9}}{\sqrt[5]{3^2}}=\frac{\sqrt[4]{3^{23}\cdot 3\cdot 3^{2\cdot 9}}}{\sqrt[5]{3^2}}=\frac{\sqrt[4]{3^{23}\cdot 3\cdot 3^{18}}}{\sqrt[5]{3^2}}=}$$

$${\displaystyle =\frac{\sqrt[4]{3^{23+1+18}}}{\sqrt[5]{3^2}}=\frac{\sqrt[4]{3^{42}}}{\sqrt[5]{3^2}}=\frac{3^{\frac{42}{4}}}{3^{\frac{2}{5}}}=3^{\frac{42}{4}-\frac{2}{5}}=3^{\frac{101}{10}}}$$

$${\displaystyle \frac{(\sqrt{6}{)}^3\cdot \sqrt[3]{6}}{\sqrt[3]{6^8}}=\frac{(6^{\frac{1}{2}}{)}^3\cdot 6^{\frac{1}{3}}}{6^{\frac{8}{6}}}=\frac{6^{\frac{3}{2}+\frac{1}{3}}}{6^{\frac{4}{3}}}=6^{\frac{3}{2}+\frac{1}{3}-\frac{4}{3}}=6^{\frac{1}{2}}}$$

Solution:

$3^{\frac{101}{10}},6^{\frac{1}{2}}$

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Development:

${\displaystyle \sqrt[4]{7^3\cdot 7^2}=\sqrt[4]{7^{3+2}}=\sqrt[4]{7^5}=7^{\frac{5}{4}}}$

${\displaystyle \sqrt[4]{{56}^8}={56}^{\frac{8}{4}}={56}^2}$

Solution:

$7^{\frac{5}{4}},{56}^2$

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