# How do you simplify 16^(-2/3)?

Apr 5, 2018

color(blue)(16^(-2/3) = 1/(4*2^(2/3

#### Explanation:

Given:
$\text{ }$
color(blue)(16^(-2/3)

Identities used:

color(red)(a^(-b)=1/(a^b)

color(red)(a^(m/n) = rootn(a^m)

Consider the expression given:

color(blue)(16^(-2/3)

$\Rightarrow \frac{1}{{16}^{\frac{2}{3}}}$

$\Rightarrow \frac{1}{\sqrt{{16}^{2}}}$

rArr 1/root3(256

$\Rightarrow \frac{1}{\sqrt{64 \cdot 4}}$

Note that color(blue)(64 = 4^3

$\Rightarrow \frac{1}{\sqrt{{4}^{3} \cdot 4}}$

$\Rightarrow \frac{1}{\sqrt{{4}^{3}}} \cdot \frac{1}{\sqrt{4}}$

Note that color(red)(rootp(m^n) = (m^n)^(1/p)

$\Rightarrow \frac{1}{{4}^{3}} ^ \left(\frac{1}{3}\right) \cdot \frac{1}{\sqrt{4}}$

$\Rightarrow \left(\frac{1}{4}\right) \cdot \frac{1}{\sqrt{4}}$

$\Rightarrow \left(\frac{1}{4}\right) \cdot \frac{1}{\sqrt{{2}^{2}}}$

Note that color(red)(sqrt(m^n) = (m^n)^(1/2)=m^(n/2)

$\Rightarrow \frac{1}{4} \cdot \left[\frac{1}{{\left({2}^{2}\right)}^{\frac{1}{3}}}\right]$

$\Rightarrow \frac{1}{4} \cdot \frac{1}{{2}^{\frac{2}{3}}}$

rArr 1/(4*2^(2/3)

Hence,

color(blue)(16^(-2/3) = 1/(4*2^(2/3)

If you wish, you can continue to simplify further:

1/(4*2^(2/3)

rArr 1/(2^2*2^(2/3)

Note that color(red)(a^m*a^n=a^(m+n)

rArr 1/2^(2+(2/3)

rArr 1/(2^(8/3)

rArr 2^(-8/3

Hope you find this solution useful.