Dividing Monomials

Simplify Expressions Using the Quotient to a Power Property

Now we will look at an example that will lead us to the Quotient to a Power Property.

	\(\left(\dfrac{x}{y}\right)^{3}\)
This means:	\(\dfrac{x}{y} \cdot \dfrac{x}{y} \cdot \dfrac{x}{y}\)
Multiply the fractions.	\(\dfrac{x \cdot x \cdot x}{y \cdot y \cdot y}\)
Write with exponents.	\(\dfrac{x^{3}}{y^{3}}\)

Notice that the exponent applies to both the numerator and the denominator.

We write:	\(\left(\dfrac{x}{y}\right)^{3}\)
	\(\dfrac{x^{3}}{y^{3}}\)

This leads to the Quotient to a Power Property for Exponents.

Quotient to a Power Property for Exponents

If \(a\) and \(b\) are real numbers, \(b \neq 0\), and \(m\) is a counting number, then

\(\left(\dfrac{a}{b}\right)^{m}=\dfrac{a^{m}}{b^{m}}\)

To raise a fraction to a power, raise the numerator and denominator to that power.

An example with numbers may help you understand this property:

\(\begin{aligned}\left(\dfrac{2}{3}\right)^{3} &=\dfrac{2^{3}}{3^{3}} \\ \dfrac{2}{3} \cdot \dfrac{2}{3} \cdot \dfrac{2}{3} &=\dfrac{8}{27} \\ \dfrac{8}{27} &=\dfrac{8}{27}\text{✓} \end{aligned}\)

Example 6.64

Simplify:

\(\left(\dfrac{3}{7}\right)^{2}\)
\(\left(\dfrac{b}{3}\right)^{4}\)
\(\left(\dfrac{k}{j}\right)^{3}\).

Solution

	\(\left(\dfrac{3}{7}\right)^{2}\)
Use the Quotient Property, \(\left(\dfrac{a}{b}\right)^{m}=\dfrac{a^{m}}{b^{m}}\).	\(\dfrac{3^{2}}{7^{2}}\)
Simplify.	\(\dfrac{9}{49}\)

	\(\left(\dfrac{b}{3}\right)^{4}\)
\(\left(\dfrac{a}{b}\right)^{m}=\dfrac{a^{m}}{b^{m}}\).	\(\dfrac{b^{4}}{3^{4}}\)
Simplify.	\(\dfrac{b^{4}}{81}\)

	\(\left(\dfrac{k}{j}\right)^{3}\)
Raise the numerator and denominator to the third power.	\(\dfrac{k^{3}}{j^{3}}\)

Try It 6.127

Simplify:

\(\left(\dfrac{5}{8}\right)^{2}\)
\(\left(\dfrac{p}{10}\right)^{4}\)
\(\left(\dfrac{m}{n}\right)^{7}\).

Try It 6.128

Simplify:

\(\left(\dfrac{1}{3}\right)^{3}\)
\(\left(\dfrac{-2}{q}\right)^{3}\)
\(\left(\dfrac{w}{x}\right)^{4}\).