## Simplifying Expressions with Exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, \({2}^{4}\) means to multiply 2 by itself 4 times, so \({2}^{4}\) means \(2·2·2·2\).

Let’s review the vocabulary for expressions with exponents.

### Exponential Notation

This is read \(a\) to the \({m}^{th}\) power.

In the expression \({a}^{m}\), the *exponent*\(m\) tells us how many times we use the *base*\(a\) as a factor.

Before we begin working with variable expressions containing exponents, let’s simplify a few expressions involving only numbers.

## Example

Simplify:

(a) \({4}^{3}\)

(b) \({7}^{1}\)

(b) \({\left(\frac{5}{6}\right)}^{2}\)

(c) \({\left(0.63\right)}^{2}.\)

Solution

(a) \(\begin{array}{cccc}& & & {4}^{3}\hfill \\ \text{Multiply three factors of 4.}\hfill & & & 4·4·4\hfill \\ \text{Simplify.}\hfill & & & 64\hfill \end{array}\)

(b) \(\begin{array}{cccc}& & & \phantom{\rule{1.1em}{0ex}}{7}^{1}\hfill \\ \text{Multiply one factor of 7.}\hfill & & & \phantom{\rule{1.1em}{0ex}}7\hfill \end{array}\)

(c) \(\begin{array}{cccc}& & & \phantom{\rule{2.2em}{0ex}}{\left(\frac{5}{6}\right)}^{2}\hfill \\ \text{Multiply two factors.}\hfill & & & \phantom{\rule{2.2em}{0ex}}\left(\frac{5}{6}\right)\left(\frac{5}{6}\right)\hfill \\ \text{Simplify.}\hfill & & & \phantom{\rule{2.2em}{0ex}}\frac{25}{36}\hfill \end{array}\)

(d) \(\begin{array}{cccc}& & & \phantom{\rule{2.2em}{0ex}}{\left(0.63\right)}^{2}\hfill \\ \text{Multiply two factors.}\hfill & & & \phantom{\rule{2.2em}{0ex}}\left(0.63\right)\left(0.63\right)\hfill \\ \text{Simplify.}\hfill & & & \phantom{\rule{2.2em}{0ex}}0.3969\hfill \end{array}\)

## Example

Simplify:

- \({\left(-5\right)}^{4}\)
- \(\text{−}{5}^{4}.\)

### Solution

\(\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}{\left(-5\right)}^{4}\hfill \\ \text{Multiply four factors of}\phantom{\rule{0.2em}{0ex}}-5.\hfill & & & \phantom{\rule{4em}{0ex}}\left(-5\right)\left(-5\right)\left(-5\right)\left(-5\right)\hfill \\ \text{Simplify.}\hfill & & & \phantom{\rule{4em}{0ex}}625\hfill \end{array}\)

\(\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}\text{−}{5}^{4}\hfill \\ \text{Multiply four factors of 5.}\hfill & & & \phantom{\rule{4em}{0ex}}\text{−}\left(5·5·5·5\right)\hfill \\ \text{Simplify.}\hfill & & & \phantom{\rule{4em}{0ex}}-625\hfill \end{array}\)

Notice the similarities and differences in both parts of the example above! Why are the answers different? As we follow the order of operations in part the parentheses tell us to raise the \(\left(-5\right)\) to the 4^{th} power. In part we raise just the 5 to the 4^{th} power and then take the opposite.