Mathematics » Introducing Polynomials » Using Multiplication Properties of Exponents

Simplifying Expressions Using the Product to a Power Property

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Simplifying Expressions Using the Product to a Power Property

We will now look at an expression containing a product that is raised to a power. Look for a pattern.

 \({\left(2x\right)}^{3}\)
What does this mean?\(2x·2x·2x\)
We group the like factors together.\(2·2·2·x·x·x\)
How many factors of 2 and of \(x?\)\({2}^{3}·{x}^{3}\)
Notice that each factor was raised to the power.\({\left(2x\right)}^{3}\phantom{\rule{0.2em}{0ex}}\text{is}\phantom{\rule{0.2em}{0ex}}{2}^{3}·{x}^{3}\)
We write:\({\left(2x\right)}^{3}\)

 

\({2}^{3}·{x}^{3}\)

The exponent applies to each of the factors. This leads to the Product to a Power Property for Exponents.

Definition: Product to a Power Property of Exponents

If \(a\) and \(b\) are real numbers and \(m\) is a whole number, then

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

To raise a product to a power, raise each factor to that power.

An example with numbers helps to verify this property:

\(\begin{array}{ccc}\hfill {\left(2·3\right)}^{2}& \stackrel{?}{=}& {2}^{2}·{3}^{2}\hfill \\ \hfill {6}^{2}& \stackrel{?}{=}& 4·9\hfill \\ \hfill 36& =& 36✓\hfill \end{array}\)

Example

Simplify: \({\left(-11x\right)}^{2}.\)

Solution

 \({\left(-11x\right)}^{2}\)
Use the Power of a Product Property, \({\left(ab\right)}^{m}={a}^{m}{b}^{m}.\)Simplifying Expressions Using the Product to a Power Property
Simplify.\(121{x}^{2}\)

Example

Simplify: \({\left(3xy\right)}^{3}.\)

Solution

 \({\left(3xy\right)}^{3}\)
Raise each factor to the third power.Simplifying Expressions Using the Product to a Power Property
Simplify.\(27{x}^{3}{y}^{3}\)

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