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Use the Law of Exponent to rewrite and simplify the
expression:

$$(a) \frac{(4)^{-3}}{(2)^{-8}}$$

$$\text { (a) } \frac{(4)^{-3}}{(2)^{-8}}=\frac{2^{8}}{4^{3}}=\frac{2^{8}}{(2^2)^{(3)}}=\frac{2^{8}}{2^{6}}$$

$$=2^{8-6}=2^{2}=4$$

(b) $$\frac{1}{\sqrt[3]{x^{4}}}$$

(b) $$\frac{1}{\sqrt[3]{x^{4}}}=\frac{1}{x^{\frac{4}{3}}}$$

$$(c) 8^{4 / 3}$$

$$\text { (c) } 8^{4 / 3}=\left(8^{4}\right)^{1 / 3}=\sqrt[3]{8^{4}}=\sqrt[3]{\left({2}^{3}\right)^{4}}$$

$$=\sqrt[3]{2^{12}}=2^{12 / 3}={2}^{4}=2 \times 2 \times 2 \times 2=16$$

$$\text { (d) } \frac{x^{2 n} \cdot x^{3 n-1}}{n^{n+2}}$$

(d) $$\frac{x^{2 n} \cdot x^{3 n-1}}{x^{n+2}}=\frac{x^{2 n+3 n-1}}{x^{n+2}}=\frac{x^{5 n-1}}{x^{n+2}}=x^{(5 n-1)-(n+2)}$$

$$={x^{5 n-1- n-2}}{x}=x^{4 n-3}$$

(e) $$\frac{\sqrt{a \sqrt{b}}}{\sqrt[3]{a b}}$$

$$=\frac{\sqrt{a\: b^{1 / 2}}}{\sqrt[3]{a} \cdot \sqrt[3]{b}}$$

$$=\frac{\left(\alpha b^{1 / 2}\right)^{1 / 2}}{a^{1 / 3} \cdot b^{1 / 3}}$$

$$=\frac{\left(a^{1 / 2}\right)\left(b^{1 / 2}\right)^{1 / 2}}{a^{1 / 3} b^{1 / 3}}$$

$$=\frac{a^{1 / 2} \cdot b^{1 / 4}}{a^{1 / 3} b^{1 / 3}}$$

$$=a^{1 / 2-1 / 3} \cdot b^{1 / 4-\frac{1}{3}}$$

$$=a^{1 / 6} \cdot b^{-1 / 12}$$

$$=\frac{a^{1 / 6}}{b^{1 / 12}}$$

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