Question

Simplify Expressions -

$\frac{\sqrt{320m{n}^5}}{\sqrt{45m^7{n}^3}}$

Short answer

The simplification of Expression will be -$\frac{8n}{3m^3}$

Step-by-step solution

Step 1.  Given Data

Expressions -

$\frac{\sqrt{320m{n}^5}}{\sqrt{45m^7{n}^3}}$

Step 2.  To Find 

Simplify The Expressions

Step 3.  Explanation 

Expressions - $\frac{\sqrt{320m{n}^5}}{\sqrt{45m^7{n}^3}}$

Neither radicand is a perfect square, so rewrite using the quotient property of square roots -

$$\begin{gathered} \frac{\sqrt{320m{n}^5}}{\sqrt{45m^7{n}^3}} \\ \sqrt{\frac{320m{n}^5}{45m^7{n}^3}} \end{gathered}$$

Step 4.  Calculation 

Remove common factors in the numerator and denominator.

$$\begin{gathered} \sqrt{\frac{320m{n}^5}{45m^7{n}^3}} \\ \sqrt{\frac{64\times 5\times m\times n^3\times n^2}{5\times 9\times m\times m^6\times n^3}} \\ \sqrt{\frac{64n^2}{9m^6}} \end{gathered}$$

simplify-

$$\begin{gathered} \sqrt{\frac{64n^2}{9m^6}} \\ \frac{8n}{3m^3} \end{gathered}$$