Write as a root: $$4^{\frac{3}{7}}, 8^{\frac{17}{5}}$$
$$4^{\frac{3}{7}}=\sqrt[7]{4^3}$$
$$8^{\frac{17}{5}}=\sqrt[5]{8^{17}}$$
$$\sqrt[7]{4^3}, \sqrt[5]{8^{17}}$$
Write as a power the following expressions:
$$\displaystyle \frac{\sqrt[4]{3^{23}\cdot 3 \cdot (3^2)^9}}{\sqrt[5]{3^2}}, \displaystyle \frac{(\sqrt{6})^3\cdot \sqrt[3]{6}}{\sqrt[3]{6^8}}$$
$$$\displaystyle \frac{\sqrt[4]{3^{23}\cdot 3 \cdot (3^2)^9}}{\sqrt[5]{3^2}}=\frac{\sqrt[4]{3^{23}\cdot 3 \cdot 3^{2\cdot 9}}}{\sqrt[5]{3^2}}=\frac{\sqrt[4]{3^{23}\cdot 3 \cdot 3^{18}}}{\sqrt[5]{3^2}}=$$$
$$$\displaystyle =\frac{\sqrt[4]{3^{23+1+18}}}{\sqrt[5]{3^2}}=\frac{\sqrt[4]{3^{42}}}{\sqrt[5]{3^2}}=\frac{3^{\frac{42}{4}}}{3^{\frac{2}{5}}}=3^{\frac{42}{4}-\frac{2}{5}}=3^{\frac{101}{10}}$$$
$$$\displaystyle \frac{(\sqrt{6})^3\cdot \sqrt[3]{6}}{\sqrt[3]{6^8}}=\frac{(6^{\frac{1}{2}})^3\cdot 6^\frac{1}{3}}{6^\frac{8}{6}}=\frac{6^{\frac{3}{2}+\frac{1}{3}}}{6^\frac{4}{3}}=6^{\frac{3}{2}+\frac{1}{3}-\frac{4}{3}}=6^\frac{1}{2}$$$
$$3^{\frac{101}{10}}, 6^\frac{1}{2}$$
Write as a power: $$\displaystyle \sqrt[4]{7^3 \cdot 7^2}$$, $$\displaystyle \sqrt[4]{56^8}$$
$$\displaystyle \sqrt[4]{7^3 \cdot 7^2}=\sqrt[4]{7^{3+2}}=\sqrt[4]{7^5}=7^{\frac{5}{4}}$$
$$\displaystyle \sqrt[4]{56^8}=56^{\frac{8}{4}}=56^2$$
$$7^{\frac{5}{4}}, 56^2$$