# How do you evaluate the expression (5^-2)^2 using the properties?

Jun 20, 2017

See a solution process below:

#### Explanation:

First, use this property of exponents to eliminate the outer exponent:

${\left({x}^{\textcolor{red}{a}}\right)}^{\textcolor{b l u e}{b}} = {x}^{\textcolor{red}{a} \times \textcolor{b l u e}{b}}$

${\left({5}^{\textcolor{red}{- 2}}\right)}^{\textcolor{b l u e}{2}} = {5}^{\textcolor{red}{- 2} \times \textcolor{b l u e}{2}} = {5}^{-} 4$

Now, use this property of exponents to eliminate the negative exponent:

${x}^{\textcolor{red}{a}} = \frac{1}{x} ^ \textcolor{red}{- a}$

${5}^{\textcolor{red}{- 4}} = \frac{1}{5} ^ \textcolor{red}{- - 4} = \frac{1}{5} ^ 4$

Now, use the definition of exponents to complete the evaluation:

$\frac{1}{5} ^ 4 = \frac{1}{5 \times 5 \times 5 \times 5} = \frac{1}{25 \times 25} = \frac{1}{625}$

Jun 20, 2017

Write the expression in an expanded form and adjust the exponents.

#### Explanation:

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

$\left({5}^{-} 2\right) \times \left({5}^{-} 2\right) = {5}^{-} 4$

${5}^{-} 4 = \frac{1}{5} ^ 4$

$\frac{1}{5} ^ 4 = \frac{1}{5 \times 5 \times 5 \times 5}$

$\frac{1}{5 \times 5 \times 5 \times 5} = \frac{1}{625}$