Mathematics » Polynomials II » Use Multiplication Properties of Exponents

Simplifying Expressions With Exponents

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 figure has two columns. In the left column is a to the m power. The m is labeled in blue as an exponent. The a is labeled in red as the base. In the right column is the text “a to the m power means multiply m factors of a.” Below this is a to the m power equals a times a times a times a, followed by an ellipsis, with “m factors” written below in blue.

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.

This figure has two columns. The left column contains 4 cubed. Below this is 4 times 4 times 4, with “3 factors” written below in blue. The right column contains negative 9 to the fifth power. Below this is negative 9 times negative 9 times negative 9 times negative 9 times negative 9, with “5 factors” written below in blue.

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:

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

Solution

  1.  

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

  2.  

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

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 4th power. In part we raise just the 5 to the 4th power and then take the opposite.

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