# How do you simplify root4 32?

Mar 8, 2016

$32 = 16 \cdot 2$
${32}^{\frac{1}{4}} = {16}^{\frac{1}{4}} \cdot {2}^{\frac{1}{4}}$
$= 2 \cdot {2}^{\frac{1}{4}}$

Mar 8, 2016

$\sqrt[4]{32} = {2}^{\text{5/4}} = 2 \sqrt[4]{2}$

#### Explanation:

We should first note that $32 = {2}^{5}$.

$= \sqrt[4]{{2}^{5}}$

To simplify this, use the rule:

$\sqrt[b]{{x}^{a}} = {x}^{a \text{/} b}$

Thus, the expression is equal to

$= {2}^{\text{5/4}}$

This is simplified. However, if you wish, there is another way to simplify it:

$= {2}^{4 \text{/"4+"1/4}}$

$= {2}^{1 + \text{1/4}}$

Use the rule:

${x}^{a + b} = {x}^{a} \left({x}^{b}\right)$

So the expression can be split up into:

$= {2}^{1} \left({2}^{\frac{1}{4}}\right)$

Which equals

$= 2 \sqrt[4]{2}$

Another way of approaching this problem is to say that $\sqrt[4]{32}$ equals

$= \sqrt[4]{{2}^{4} \cdot 2}$

The ${2}^{4}$ can be brought out of the root:

$= 2 \sqrt[4]{2}$