# Square Roots of Products and Fractions

Whenever you need to take the square root of a big number, you can always factorize (rewrite a number as other numbers multiplied together) the number and instead take the square root of its factors, one by one. When you do this, it can be useful to look for ways to factorize your big number into a square number multiplied by something else. In that case, the calculation becomes much simpler. Have a look at the examples in the boxes below.

Rule

### TheSquareRootofaProduct

 $\sqrt{a\cdot b}=\sqrt{a}\cdot \sqrt{b}\phantom{\rule{0.33em}{0ex}}\phantom{\rule{0.33em}{0ex}}\phantom{\rule{0.33em}{0ex}}for\phantom{\rule{0.33em}{0ex}}a,b\ge 0$

Example 1

Find the square root of $32$.

 $\sqrt{32}=\sqrt{16\cdot 2}=\sqrt{16}\cdot \sqrt{2}=4\sqrt{2}$

I could have chosen to factorize $32=4\cdot 8$, but taking the square root of those factors wouldn’t help much. $32=16\cdot 2$ is a factorization including a square number, and that makes the calculation much simpler. I leave the answer as $4\sqrt{2}$, because $\sqrt{2}$ is an exact number while its decimal form will always be an approximation.

Example 2

Find the square root of $50$.

 $\sqrt{50}=\sqrt{25\cdot 2}=\sqrt{25}\cdot \sqrt{2}=5\sqrt{2}$

Here, you first have to factorize $50$ before you take the square root of each factor.

Example 3

Find the square root of $72$.

$\begin{array}{llll}\hfill \sqrt{72}& =\sqrt{9\cdot 8}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\sqrt{9\cdot 4\cdot 2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\sqrt{9}\cdot \sqrt{4}\cdot \sqrt{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =3\cdot 2\cdot \sqrt{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =6\cdot \sqrt{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$ You first factorize $72=9\cdot 4\cdot 2$. You know the answer to $\sqrt{9}$ and $\sqrt{4}$. $\sqrt{2}$ becomes a never ending decimal number, so you can just leave it as is in the final answer.

Rule

### Squarerootofafraction

 $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$

Whenever you want to take the square root a fraction, you can instead take the square root of the numerator and the denominator by themselves.

I’ve said it before and I’ll say it again: You have to memorize the formulas. You won’t stand a chance without them. Be in full control of the square root of a product and a fraction. Here are a couple of examples.

Example 4

Find the square root of $\frac{25}{16}$.

 $\sqrt{\frac{25}{16}}=\frac{\sqrt{25}}{\sqrt{16}}=\frac{5}{4}$

Example 5

Find the square root of $\frac{361}{169}$.

 $\sqrt{\frac{361}{169}}=\frac{\sqrt{361}}{\sqrt{169}}=\frac{19}{13}$

If you want to practice factorization in general, I recommend that you watch instructional videos about factorization.