# Problems from Powers of fractional exponent

Write as a root: $$4^{\frac{3}{7}}, 8^{\frac{17}{5}}$$

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### 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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Write as a power the following expressions:

$$\displaystyle \frac{\sqrt[4]{3^{23}\cdot 3 \cdot (3^2)^9}}{\sqrt[5]{3^2}}, \displaystyle \frac{(\sqrt{6})^3\cdot \sqrt[3]{6}}{\sqrt[3]{6^8}}$$

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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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Write as a power: $$\displaystyle \sqrt[4]{7^3 \cdot 7^2}$$, $$\displaystyle \sqrt[4]{56^8}$$

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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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