# How do you simplify (5^-2)^-3?

Jun 22, 2016

${\left({5}^{- 2}\right)}^{- 3} = {5}^{6}$

#### Explanation:

As ${\left({a}^{m}\right)}^{n} = {a}^{m \times n}$

${\left({5}^{- 2}\right)}^{- 3}$

= ${5}^{\left(- 2\right) \times \left(- 3\right)}$

= ${5}^{6}$

Jun 23, 2016

${5}^{6}$
With practice you could solve this in 1 to 2 lines. I have used a lot more than that to explain things.

#### Explanation:

color(blue)("Step 1")" "Dealing with just the brackets,

Consider just ${5}^{- 2}$

This is stating that 5 is raised to the power of 2 but because we have negative 2 we have to invert it ( turn upside down ).

So write ${5}^{- 2} \text{ as } {5}^{- 2} / 1$

Note that writing as ${5}^{- 2} / 1$ is not normally done but it is correct.

Inverting gives:

$\frac{1}{5} ^ \left(2\right) \leftarrow \text{ Notice that the negative( minus) sign of -2 is now +2}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Step 2}}$

We now have: ${\left(\frac{1}{5} ^ 2\right)}^{- 3}$

Notice that both the numerator and denominator have this index applied.

Inverting

${\left({5}^{2} / 1\right)}^{3} \leftarrow \text{ The negative( minus) sign of -3 is now +3}$

$\frac{{\left({5}^{2}\right)}^{3}}{1} ^ 3 = {5}^{2 \times 3} = {5}^{6}$