# How do you evaluate square root of 32a^8b + square root of 50a^16b?

Dec 31, 2015

Use properties of square root of products to find:

$\sqrt{32 {a}^{8} b} + \sqrt{50 {a}^{16} b} = {a}^{4} \left(4 + 5 {a}^{4}\right) \sqrt{2 b}$

#### Explanation:

If $x \ge 0$ or $y \ge 0$, then $\sqrt{x y} = \sqrt{x} \sqrt{y}$

(Note this does not hold if both $x < 0$ and $y < 0$)

So we find:

$\sqrt{32 {a}^{8} b} + \sqrt{50 {a}^{16} b}$

$= \sqrt{16 {a}^{8}} \sqrt{2 b} + \sqrt{25 {a}^{16}} \sqrt{2 b}$

$= \sqrt{{\left(4 {a}^{4}\right)}^{2}} \sqrt{2 b} + \sqrt{{\left(5 {a}^{8}\right)}^{2}} \sqrt{2 b}$

$= \left(4 {a}^{4} + 5 {a}^{8}\right) \sqrt{2 b}$

$= {a}^{4} \left(4 + 5 {a}^{4}\right) \sqrt{2 b}$