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Simplifying Expressions Using the Quotient to a Power Property

Simplifying 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.

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

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

We see that \({\left(\frac{x}{y}\right)}^{3}\) is \(\frac{{x}^{3}}{{y}^{3}}.\)

\(\begin{array}{ccccc}\text{We write:}\hfill & & & & {\left(\frac{x}{y}\right)}^{3}\hfill \\ & & & & \frac{{x}^{3}}{{y}^{3}}\hfill \end{array}\)

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

Definition: Quotient to a Power Property of Exponents

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

\({\left(\frac{a}{b}\right)}^{m}=\phantom{\rule{0.2em}{0ex}}\frac{{a}^{m}}{{b}^{m}}\)

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:

\(\begin{array}{ccc}\hfill {\left(\frac{2}{3}\right)}^{3}& \stackrel{?}{=}& \frac{{2}^{3}}{{3}^{3}}\hfill \\ \hfill \frac{2}{3}\cdot \frac{2}{3}\cdot \frac{2}{3}& \stackrel{?}{=}& \frac{8}{27}\hfill \\ \hfill \frac{8}{27}& =& \frac{8}{27}✓\hfill \end{array}\)

Example

Simplify:

  1. \(\phantom{\rule{0.2em}{0ex}}{\left(\frac{5}{8}\right)}^{2}\)
  2. \(\phantom{\rule{0.2em}{0ex}}{\left(\frac{x}{3}\right)}^{4}\)
  3. \(\phantom{\rule{0.2em}{0ex}}{\left(\frac{y}{m}\right)}^{3}\)

Solution

  
 Simplifying Expressions Using the Quotient to a Power Property
Use the Quotient to a Power Property, \({\left(\frac{a}{b}\right)}^{m}=\phantom{\rule{0.2em}{0ex}}\frac{{a}^{m}}{{b}^{m}}\).Simplifying Expressions Using the Quotient to a Power Property
Simplify.Simplifying Expressions Using the Quotient to a Power Property
  
 Simplifying Expressions Using the Quotient to a Power Property
Use the Quotient to a Power Property, \({\left(\frac{a}{b}\right)}^{m}=\phantom{\rule{0.2em}{0ex}}\frac{{a}^{m}}{{b}^{m}}\).Simplifying Expressions Using the Quotient to a Power Property
Simplify.Simplifying Expressions Using the Quotient to a Power Property
  
 Simplifying Expressions Using the Quotient to a Power Property
Raise the numerator and denominator to the third power.Simplifying Expressions Using the Quotient to a Power Property

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