Simplify 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. Can you find this pattern?
\(\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}{\left(2x\right)}^{3}\hfill \\ \text{What does this mean?}\hfill & & & \phantom{\rule{4em}{0ex}}2x·2x·2x\hfill \\ \text{We group the like factors together.}\hfill & & & \phantom{\rule{4em}{0ex}}2·2·2·x·x·x\hfill \\ \text{How many factors of 2 and of}\phantom{\rule{0.2em}{0ex}}x?\hfill & & & \phantom{\rule{4em}{0ex}}{2}^{3}·{x}^{3}\hfill \end{array}\)
Notice that each factor was raised to the power and \({\left(2x\right)}^{3}\) is \({2}^{3}·{x}^{3}\).
\(\begin{array}{cccc}\text{We write:}\hfill & & & \phantom{\rule{4em}{0ex}}{\left(2x\right)}^{3}\hfill \\ & & & \phantom{\rule{4em}{0ex}}{2}^{3}·{x}^{3}\hfill \end{array}\)
The exponent applies to each of the factors! This leads to the Product to a Power Property for Exponents.
Product to a Power Property for Exponents
If \(a\) and \(b\) are real numbers and \(m\) is a whole number, then
To raise a product to a power, raise each factor to that power.
An example with numbers helps to verify this property:
\(\begin{array}{ccc}\hfill {\left(2·3\right)}^{2}& \stackrel{?}{=}\hfill & {2}^{2}·{3}^{2}\hfill \\ \hfill {6}^{2}& \stackrel{?}{=}\hfill & 4·9\hfill \\ \hfill 36& =\hfill & 36\phantom{\rule{0.2em}{0ex}}\text{✓}\hfill \end{array}\)
Example
Simplify:
- \({\left(-9d\right)}^{2}\)
- \({\left(3mn\right)}^{3}.\)
Solution
-
Use Power of a Product Property, (ab)m = ambm. Simplify. -
Use Power of a Product Property, (ab)m = ambm. Simplify.
This lesson is part of:
Polynomials II