# How do you simplify sqrt(80)?

Jul 23, 2015

If $a , b \ge 0$ then $\sqrt{a b} = \sqrt{a} \sqrt{b}$
If $a \ge 0$ then $\sqrt{{a}^{2}} = a$

Hence $\sqrt{80} = \sqrt{16 \cdot 5} = 4 \sqrt{5}$

#### Explanation:

$\sqrt{80} = \sqrt{16 \cdot 5} = \sqrt{{4}^{2} \cdot 5} = \sqrt{{4}^{2}} \sqrt{5} = 4 \sqrt{5}$

Note that since $\sqrt{81} = \sqrt{{9}^{2}} = 9$ then

$\frac{9}{4} = \frac{\sqrt{81}}{\frac{\sqrt{80}}{\sqrt{5}}} = \sqrt{\frac{81}{80}} \sqrt{5}$ is a useful approximation for $\sqrt{5}$.