# How do you write 5^-6 with positive exponents?

Aug 26, 2016

${5}^{-} 6 = \frac{1}{5} ^ 6$

#### Explanation:

One of the laws of indices states:

${x}^{\textcolor{red}{\left(- m\right)}} = \frac{1}{x} ^ \textcolor{red}{\left(+ m\right)}$

We can see that when the base of $x$ was moved from the numerator to the denominator, the sign of the index changed.

The same will happen if a base is moved from the denominator to the numerator.

$\frac{1}{x} ^ \textcolor{b l u e}{\left(- n\right)} = {x}^{\textcolor{b l u e}{\left(+ n\right)}}$

${5}^{-} 6 = \frac{1}{5} ^ 6$

Note:

• This is better written in this form, rather than $\frac{1}{15625}$

• This has nothing to do with the negative numbers like $- 5 \mathmr{and} - 8$ on the number line!

• Only the base with the negative index is moved. All the other numbers and bases stay the same.

$2 {x}^{3} \textcolor{red}{{y}^{-} 4} = \frac{2 {x}^{3}}{\textcolor{red}{{y}^{4}}}$