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Simplifying Expressions Using the Power Property For Exponents

Simplifying Expressions Using the Power Property For Exponents

Now let’s look at an exponential expression that contains a power raised to a power. See if you can discover a general property.

 x squared, in parentheses, cubed.
What does this mean?
How many factors altogether?
x squared cubed is x squared times x squared times x squared, which is x times x, multiplied by x times x, multiplied by x times x. x times x has two factors. Two plus two plus two is six factors.
So we havex to the sixth power.
Notice that 6 is the product of the exponents, 2 and 3.x squared cubed is x to the power of 2 times 3, or x to the sixth power.

We write:

\(\begin{array}{c}\hfill {\left({x}^{2}\right)}^{3}\hfill \\ \hfill {x}^{2·3}\hfill \\ \hfill {x}^{6}\hfill \end{array}\)

We multiplied the exponents. This leads to the Power Property for Exponents.

Power Property for Exponents

If \(a\) is a real number, and \(m\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n\) are whole numbers, then

\({\left({a}^{m}\right)}^{n}={a}^{m·n}\)

To raise a power to a power, multiply the exponents.

An example with numbers helps to verify this property.

\(\begin{array}{ccc}\hfill {\left({3}^{2}\right)}^{3}& \stackrel{?}{=}\hfill & {3}^{2·3}\hfill \\ \hfill {\left(9\right)}^{3}& \stackrel{?}{=}\hfill & {3}^{6}\hfill \\ \hfill 729& =\hfill & 729\phantom{\rule{0.2em}{0ex}}\text{✓}\hfill \end{array}\)

Example

Simplify: (a) \({\left({y}^{5}\right)}^{9}\) (b) \({\left({4}^{4}\right)}^{7}.\)

Solution

(a)
 y to the fifth power, in parentheses, to the ninth power.
Use the power property, (am)n = am·n.y to the power of 5 times 9.
Simplify.y to the 45th power.
(b)
 4 to the fourth power, in parentheses, to the 7th power.
Use the power property.4 to the power of 4 times 7.
Simplify.4 to the twenty-eighth power.

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