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Examples of Solving Powers with Variables

The six rules of powers from the previous entries are useful in many cases. If you understand these crucial rules, you’ll be on the right track to mastering the mathematics that lies ahead.

Here are some examples combining all the rules. Make sure you understand them!

Example 1

Write ${(x y)}^{3} \cdot x^{2} y^{- 3}$ as simply as possible

\begin{array}{l} {(x y)}^{3} \cdot x^{2} y^{- 3} & = x^{3} y^{3} x^{2} y^{- 3} \\ = x^{3 + 2} y^{3 - 3} \\ = x^{5} y^{0} \\ = x^{5} \cdot 1 \\ = x^{5} \end{array}

Example 2

Write ${(\frac{x}{y})}^{- 3}$ as simply as possible

{(\frac{x}{y})}^{- 3} = \frac{x^{- 3}}{y^{- 3}} = \frac{1}{x^{3} y^{- 3}} = \frac{y^{3}}{x^{3}}

Note that you can actually just flip the fraction and change the sign of the exponent, like this:

{(\frac{x}{y})}^{- 3} = {(\frac{y}{x})}^{3} = \frac{y^{3}}{x^{3}}

Example 3

Write $3^{3} a^{- 3} b {(a b)}^{3}$ as simply as possible

\begin{array}{l} 3^{3} a^{- 3} b {(a b)}^{3} & = 27 a^{- 3} b a^{3} b^{3} \\ = 27 a^{- 3 + 3} b^{1 + 3} \\ = 27 a^{0} b^{4} \\ = 27 b^{4} \end{array}

Example 4

Write $\frac{2 x^{2} y^{2}}{2 x y}$ as simply as possible

\begin{array}{l} \frac{2 x^{2} y^{2}}{2 x y} & = \frac{2^{1} x^{2} y^{2}}{2^{1} x^{1} y^{1}} \\ = \frac{2^{1} 2^{- 1} x^{2} x^{- 1} y^{2} y^{- 1}}{1} \\ = 2^{1 - 1} x^{2 - 1} y^{2 - 1} \\ = 2^{0} x^{1} y^{1} \\ = x y \end{array}

Example 5

Write $\frac{a^{2} b^{- 4} 2^{2}}{{(2 a b)}^{- 3}}$ as simply as possible

\begin{array}{l} \frac{a^{2} b^{- 4} 2^{2}}{{(2 a b)}^{- 3}} & = \frac{a^{2} b^{- 4} 2^{2}}{2^{- 3} a^{- 3} b^{- 3}} \\ = a^{2} a^{3} b^{- 4} b^{3} 2^{2} 2^{3} \\ = a^{2 + 3} b^{- 4 + 3} 2^{2 + 3} \\ = a^{5} b^{- 1} 2^{5} \\ = \frac{2^{5} a^{5}}{b} \end{array}

How to Multiply Powers in Parentheses

What Is the Connection Between Powers and Roots?

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