# How do you simplify (1/8)^(4/3)?

Aug 20, 2016

$\frac{1}{16}$

#### Explanation:

First, we see that:

$\frac{1}{8} = \frac{1}{2} ^ 3 = {2}^{-} 3$

Thus:

${\left(\frac{1}{8}\right)}^{\frac{4}{3}} = {\left({2}^{-} 3\right)}^{\frac{4}{3}}$

Now use the rule ${\left({a}^{b}\right)}^{c} = {a}^{b c}$:

${\left({2}^{-} 3\right)}^{\frac{4}{3}} = {2}^{- 3 \times \frac{4}{3}} = {2}^{-} 4 = \frac{1}{2} ^ 4 = \frac{1}{16}$

Aug 24, 2016

$\frac{1}{16}$

#### Explanation:

One of the laws of indices states:

$\sqrt[\textcolor{red}{q}]{{x}^{\textcolor{b l u e}{p}}} = {\left(\sqrt[\textcolor{red}{q}]{x}\right)}^{\textcolor{b l u e}{p}} = {x}^{\frac{\textcolor{b l u e}{p}}{\textcolor{red}{q}}}$

The index is a fraction: note that:

The denominator indicates the root and the numerator the power.
It does not matter which you do first, but finding the root first means we can use smaller numbers, rather than finding ${8}^{4}$

${\left(\frac{1}{8}\right)}^{\frac{4}{3}}$ can be written as (color(red)(root3(1/8)))^color(blue)(4)color(white)(xxxxxxxxxxxx) "8 = 2^3

= ${\textcolor{red}{\left(\frac{1}{2}\right)}}^{\textcolor{b l u e}{4}}$

=$\frac{1}{{2}^{4}} \textcolor{w h i t e}{\times \times \times \times \times \times \times \times \times \times \times \times \times \times \times} {1}^{x} = 1$

=$\frac{1}{16}$

It will be to your advantage to learn the powers of the natural numbers from 1 to 10 up to 1000.

In this question we have used ${2}^{3} = 8 \mathmr{and} {2}^{4} = 16$