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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.

\(\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}{\left(\cfrac{x}{y}\right)}^{3}\hfill \\ \text{This means:}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\cfrac{x}{y}·\cfrac{x}{y}·\cfrac{x}{y}\hfill \\ \text{Multiply the fractions.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\cfrac{x·x·x}{y·y·y}\hfill \\ \text{Write with exponents.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\cfrac{{x}^{3}}{{y}^{3}}\hfill \end{array}\)

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

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

\(\begin{array}{cccc}\text{We write:}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}{\left(\cfrac{x}{y}\right)}^{3}\hfill \\ & & & \hfill \phantom{\rule{4em}{0ex}}\cfrac{{x}^{3}}{{y}^{3}}\hfill \end{array}\)

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, \(b\ne 0\), and \(m\) is a counting number, then

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

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:

\(\begin{array}{ccc}\hfill {\left(\cfrac{2}{3}\right)}^{3}& =\hfill & \cfrac{{2}^{3}}{{3}^{3}}\hfill \\ \hfill \cfrac{2}{3}·\cfrac{2}{3}·\cfrac{2}{3}& =\hfill & \cfrac{8}{27}\hfill \\ \hfill \cfrac{8}{27}& =\hfill & \cfrac{8}{27}✓\hfill \end{array}\)

Example

Simplify:

(a) \({\left(\cfrac{3}{7}\right)}^{2}\)

(b) \({\left(\cfrac{b}{3}\right)}^{4}\)

(c) \({\left(\cfrac{k}{j}\right)}^{3}.\)

Solution

(a)
 3 sevenths squared.
Use the Quotient Property, \({\left(\cfrac{a}{b}\right)}^{m}=\cfrac{{a}^{m}}{{b}^{m}}\).3 squared divided by 7 squared.
Simplify.9 forty-ninths.
(b)
 b thirds to the fourth power.
Use the Quotient Property, \({\left(\cfrac{a}{b}\right)}^{m}=\cfrac{{a}^{m}}{{b}^{m}}\).b to the fourth power divided by 3 to the fourth power.
Simplify.b to the fourth power divided by 81.
(c)
 k divided by j, in parentheses, cubed.
Raise the numerator and denominator to the third power.k cubed divided by j cubed.

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