# Question #45348

Mar 17, 2017

${16}^{\frac{2}{3}}$ can be rewritten as $\sqrt[3]{{16}^{2}}$

Solve for the inside first:

${16}^{2} = 256$

Now take the third root of this. Break down 256 into powers of 2:

$256 = {2}^{8}$

$8$ goes into $3$ 2 times and you have a remainder of ${2}^{2} = 4$ So we have:

$2 \cdot 2 \sqrt[3]{4}$

Mar 17, 2017

$4 \sqrt[3]{{2}^{2}}$

#### Explanation:

Before you do anything else it is wise to look and see if you can spot any short cuts. I have spotted 1

Just accept that ${16}^{\frac{2}{3}}$ is another way of writing $\sqrt[3]{{16}^{2}}$

When ever you have a root check to see if there are any values you can 'extract' from it. In this case we would be looking for cubed value.

$16 = {4}^{2} = {2}^{2} \times {2}^{2} = {2}^{3} \times 2$ so

${16}^{2} = {2}^{3} \times 2 \times {2}^{3} \times 2 = {2}^{3} \times {2}^{3} \times {2}^{2}$

$\sqrt[3]{{16}^{2}} \to \sqrt[3]{{2}^{3} \times {2}^{3} \times {2}^{2}} \text{ "=" "2xx2xxroot(3)(2^2)" "=" } 4 \sqrt[3]{{2}^{2}}$