# How do you simplify sqrt24*sqrt(80/192)?

Jul 17, 2017

See a solution process below:

#### Explanation:

First, rewrite this expression as:

$\sqrt{4 \cdot 6} \sqrt{\frac{16 \cdot 5}{64 \cdot 3}}$

We can then use these rules for radicals to begin simplifying the expression:

$\sqrt{\frac{\textcolor{red}{a}}{\textcolor{b l u e}{b}}} = \frac{\sqrt{\textcolor{red}{a}}}{\sqrt{\textcolor{b l u e}{b}}}$ and $\sqrt{\textcolor{red}{a} \cdot \textcolor{b l u e}{b}} = \sqrt{\textcolor{red}{a}} \cdot \sqrt{\textcolor{b l u e}{b}}$

$\sqrt{4 \cdot 6} \sqrt{\frac{16 \cdot 5}{64 \cdot 3}} \implies$

$\sqrt{4 \cdot 6} \frac{\sqrt{\left(16 \cdot 5\right)}}{\sqrt{\left(64 \cdot 3\right)}} \implies$

$\sqrt{4} \sqrt{6} \frac{\sqrt{16} \sqrt{5}}{\sqrt{64} \sqrt{3}} \implies$

$2 \sqrt{6} \frac{4 \sqrt{5}}{8 \sqrt{3}} \implies$

$\left(2 \cdot \frac{4}{8}\right) \sqrt{6} \frac{\sqrt{5}}{\sqrt{3}} \implies$

$\left(\frac{8}{8}\right) \sqrt{6} \frac{\sqrt{5}}{\sqrt{3}} \implies$

$1 \sqrt{6} \frac{\sqrt{5}}{\sqrt{3}} \implies$

$\sqrt{6} \frac{\sqrt{5}}{\sqrt{3}}$

We can now use the reverse of the rules above to complete the simplification process:

$\sqrt{6} \frac{\sqrt{5}}{\sqrt{3}} \implies$

$\frac{\sqrt{6} \sqrt{5}}{\sqrt{3}} \implies$

$\frac{\sqrt{6 \cdot 5}}{\sqrt{3}} \implies$

$\sqrt{\frac{6 \cdot 5}{3}} \implies$

$\sqrt{\frac{30}{3}} \implies$

$\sqrt{10}$

Jul 17, 2017

$\sqrt{10}$

#### Explanation:

Square roots which are multiplied or divided can be combined into one square root:

$\sqrt{24} \times \sqrt{\frac{80}{192}} = \sqrt{\frac{24 \times 80}{192}}$

Write each number as the product of its factors, using square numbers where possible.

sqrt((24xx80)/192) =sqrt((6xx4xx16xx5)/(3xx4xx16)

Simplify:
sqrt((cancel6^2xxcancel4xxcancel16xx5)/(cancel3xxcancel4xxcancel16)

$= \sqrt{10}$

If there were no square numbers, you would simply have used the prime factors which would give the same results