Question

write the expression in simplest form. assume all variables are positive. $sqrt5{\frac{160m^{6}}{n^{7}}} = \frac{square}{square}$

Explanation:

Step1: Split root into product/quotient

$$\sqrt[5]{\frac{160m^6}{n^7}} = \frac{\sqrt[5]{160m^6}}{\sqrt[5]{n^7}}$$

Step2: Factor perfect 5th powers

$$\frac{\sqrt[5]{32 \cdot 5 \cdot m^5 \cdot m}}{\sqrt[5]{n^5 \cdot n^2}}$$

Step3: Simplify each root

$$\frac{\sqrt[5]{32}\cdot\sqrt[5]{m^5}\cdot\sqrt[5]{5m}}{\sqrt[5]{n^5}\cdot\sqrt[5]{n^2}} = \frac{2m\sqrt[5]{5m}}{n\sqrt[5]{n^2}}$$

Step4: Rationalize the denominator

$$\frac{2m\sqrt[5]{5m} \cdot \sqrt[5]{n^3}}{n\sqrt[5]{n^2} \cdot \sqrt[5]{n^3}} = \frac{2m\sqrt[5]{5mn^3}}{n\sqrt[5]{n^5}}$$

Step5: Simplify the denominator root

$$\frac{2m\sqrt[5]{5mn^3}}{n \cdot n} = \frac{2m\sqrt[5]{5mn^3}}{n^2}$$

Answer:

$\frac{2m\sqrt[5]{5mn^3}}{n^2}$