Mathematics » Introducing Polynomials » Using Multiplication Properties of Exponents

Simplifying Expressions Using the Product to a Power Property

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.\(27{x}^{3}{y}^{3}\)

[Attributions and Licenses]


This is a lesson from the tutorial, Introducing Polynomials and you are encouraged to log in or register, so that you can track your progress.

Log In

Discussion


Do NOT follow this link or you will be banned from the site!