# How do you evaluate the expression (1/5)^-4/((1/5)^-2(1/5)^-5) using the properties?

##### 1 Answer
May 25, 2017

$\frac{1}{125}$

#### Explanation:

Set $x = \frac{1}{5}$ to make things clearer

${x}^{- 4} / \left({x}^{- 2} {x}^{- 5}\right)$

$= {x}^{- 4} / {x}^{- 7}$

Note that $\frac{1}{x} ^ \left(- 7\right)$ is the same as ${x}^{7}$

Note that ${x}^{- 4}$ is the same as $\frac{1}{x} ^ 4$

Putting it all together we have:

${x}^{7} / {x}^{4}$
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Method 1: $\to {x}^{7 - 4} = {x}^{3}$

Method 2: $\to \frac{1}{\cancel{{x}^{4}}} \times \cancel{{x}^{4}} \times {x}^{3} \text{ "=" } {x}^{3}$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Finally, replace $x$ with $\frac{1}{5}$ and simplify.

${x}^{3} = {\left(\frac{1}{5}\right)}^{3} = {1}^{3} / {5}^{3} = \frac{1}{125}$

Final Answer