Mathematics » Introducing Polynomials » Using Multiplication Properties of Exponents

Simplifying Expressions Using the Power Property of Exponents Continued

Simplifying Expressions Using the Power Property of Exponents Continued

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

 Simplifying Expressions Using the Power Property of Exponents Continued
 Simplifying Expressions Using the Power Property of Exponents Continued
What does this mean?

 
 

How many factors altogether?

Simplifying Expressions Using the Power Property of Exponents Continued
So, we haveSimplifying Expressions Using the Power Property of Exponents Continued
Notice that 6 is the product of the exponents, 2 and 3.Simplifying Expressions Using the Power Property of Exponents Continued
We write:\({\left({x}^{2}\right)}^{3}\)

 

\({x}^{2\cdot 3}\)

 

\({x}^{6}\)

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

Definition: Power Property of Exponents

If \(a\) is a real number and \(m,n\) 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.

\(\begin{array}{ccc}\hfill {\left({5}^{2}\right)}^{3}& \stackrel{?}{=}& {5}^{2·3}\hfill \\ \hfill {\left(25\right)}^{3}& \stackrel{?}{=}& {5}^{6}\hfill \\ \hfill 15,625& =& 15,625✓\hfill \end{array}\)

Example

Simplify:

  1. \(\phantom{\rule{0.2em}{0ex}}{\left({x}^{5}\right)}^{7}\)
  2. \(\phantom{\rule{0.2em}{0ex}}{\left({3}^{6}\right)}^{8}\)

Solution

  
 \({\left({x}^{5}\right)}^{7}\)
Use the Power Property, \({\left({a}^{m}\right)}^{n}={a}^{m·n}.\)Simplifying Expressions Using the Power Property of Exponents Continued
Simplify.\({x}^{35}\)
  
 \({\left({3}^{6}\right)}^{8}\)
Use the Power Property, \({\left({a}^{m}\right)}^{n}={a}^{m·n}.\)Simplifying Expressions Using the Power Property of Exponents Continued
Simplify.\({3}^{48}\)

Optional Video: Exponent Properties II

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