6.5 Divide monomials

Simplify expressions using the quotient property for exponents

Earlier in this chapter, we developed the properties of exponents for multiplication. We summarize these properties below.

Now we will look at the exponent properties for division. A quick memory refresher may help before we get started. You have learned to simplify fractions by dividing out common factors from the numerator and denominator using the Equivalent Fractions Property. This property will also help you work with algebraic fractions—which are also quotients.

As before, we’ll try to discover a property by looking at some examples.

$\begin{array}{cccccccccc}\text{Consider}\hfill & & & \hfill \frac{x^5}{x^2}\hfill & & & \hfill \text{and}\hfill & & & \hfill \frac{x^2}{x^3}\hfill \\ \text{What do they mean?}\hfill & & & \hfill \frac{x·x·x·x·x}{x·x}\hfill & & & & & & \hfill \frac{x·x}{x·x·x}\hfill \\ \text{Use the Equivalent Fractions Property.}\hfill & & & \hfill \frac{\cancel{x}·\cancel{x}·x·x·x}{\cancel{x}·\cancel{x}}\hfill & & & & & & \hfill \frac{\cancel{x}·\cancel{x}·1}{\cancel{x}·\cancel{x}·x}\hfill \\ \text{Simplify.}\hfill & & & \hfill x^3\hfill & & & & & & \hfill \frac{1}{x}\hfill \end{array}$

Notice, in each case the bases were the same and we subtracted exponents.

When the larger exponent was in the numerator, we were left with factors in the numerator.

When the larger exponent was in the denominator, we were left with factors in the denominator—notice the numerator of 1.

We write:

This leads to the Quotient Property for Exponents .

A couple of examples with numbers may help to verify this property.