Learning Outcomes
- Use the product property of exponents to simplify expressions
- Use the power property of exponents to simplify expressions
- Use the product to a power property of exponents to simplify expressions
You have seen that when you combine like terms by adding and subtracting, you need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too. We’ll derive the properties of exponents by looking for patterns in several examples. All the exponent properties hold true for any real numbers, but right now we will only use whole number exponents.
First, we will look at an example that leads to the Product Property.
| [latex]{x}^{2}\cdot{x}^{3}[/latex] | |
| What does this mean?
How many factors altogether? |
 |
| So, we have | [latex]{x}\cdot{x}\cdot{x}\cdot{x}\cdot{x}={x}^{5}[/latex] |
| Notice that [latex]5[/latex] is the sum of the exponents, [latex]2[/latex] and [latex]3[/latex]. | [latex]{x}^{2}\cdot{x}^{3}[/latex] is [latex]{x}^{2+3}[/latex], or [latex]{x}^{5}[/latex] |
| We write: | [latex]{x}^{2}\cdot {x}^{3}[/latex]
[latex]{x}^{2+3}[/latex] [latex]{x}^{5}[/latex] |
The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.
Product Property of Exponents
If [latex]a[/latex] is a real number and [latex]m,n[/latex] are counting numbers, then
[latex]{a}^{m}\cdot {a}^{n}={a}^{m+n}[/latex]
To multiply with like bases, add the exponents.
An example with numbers helps to verify this property.
[latex]\begin{array}{ccc}\hfill {2}^{2}\cdot {2}^{3}& \stackrel{?}{=}& {2}^{2+3}\hfill \\ \hfill 4\cdot 8& \stackrel{?}{=}& {2}^{5}\hfill \\ \hfill 32& =& 32\hfill \end{array}[/latex]
example
Simplify: [latex]{x}^{5}\cdot {x}^{7}[/latex]
Solution
| [latex]{x}^{5}\cdot {x}^{7}[/latex] | |
| Use the product property, [latex]{a}^{m}\cdot {a}^{n}={a}^{m+n}[/latex]. | [latex]x^{\color{red}{5+7}}[/latex] |
| Simplify. | [latex]{x}^{12}[/latex] |
try it
example
Simplify: [latex]{b}^{4}\cdot b[/latex]
try it
example
Simplify: [latex]{2}^{7}\cdot {2}^{9}[/latex]
try it
example
Simplify: [latex]{y}^{17}\cdot {y}^{23}[/latex]
try it
We can extend the Product Property of Exponents to more than two factors.
example
Simplify: [latex]{x}^{3}\cdot {x}^{4}\cdot {x}^{2}[/latex]
try it
In the following video we show more examples of how to use the product rule for exponents to simplify expressions.
Simplify Expressions Using the Power Property of 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.
| [latex]({x}^{2})^{3}[/latex] | |
| [latex]{x}^{2}\cdot{x}^{2}\cdot{x}^{2}[/latex] | |
| What does this mean?
How many factors altogether? |
 |
| So, we have | [latex]{x}\cdot{x}\cdot{x}\cdot{x}\cdot{x}\cdot{x}={x}^{6}[/latex] |
| Notice that [latex]6[/latex] is the product of the exponents, [latex]2[/latex] and [latex]3[/latex]. | [latex]({x}^{2})^{3}[/latex] is [latex]{x}^{2\cdot3}[/latex] or [latex]{x}^{6}[/latex] |
| We write: | [latex]{\left({x}^{2}\right)}^{3}[/latex]
[latex]{x}^{2\cdot 3}[/latex] [latex]{x}^{6}[/latex] |
We multiplied the exponents. This leads to the Power Property for Exponents.
Power Property of Exponents
If [latex]a[/latex] is a real number and [latex]m,n[/latex] are whole numbers, then
[latex]{\left({a}^{m}\right)}^{n}={a}^{m\cdot n}[/latex]
To raise a power to a power, multiply the exponents.
An example with numbers helps to verify this property.
[latex]\begin{array}{ccc}\hfill {\left({5}^{2}\right)}^{3}& \stackrel{?}{=}& {5}^{2\cdot 3}\hfill \\ \hfill {\left(25\right)}^{3}& \stackrel{?}{=}& {5}^{6}\hfill \\ \hfill 15,625& =& 15,625\hfill \end{array}[/latex]
example
Simplify:
1. [latex]{\left({x}^{5}\right)}^{7}[/latex]
2. [latex]{\left({3}^{6}\right)}^{8}[/latex]
try it
Watch the following video to see more examples of how to use the power rule for exponents to simplify expressions.
Simplify 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.
| [latex]{\left(2x\right)}^{3}[/latex] | |
| What does this mean? | [latex]2x\cdot 2x\cdot 2x[/latex] |
| We group the like factors together. | [latex]2\cdot 2\cdot 2\cdot x\cdot x\cdot x[/latex] |
| How many factors of [latex]2[/latex] and of [latex]x?[/latex] | [latex]{2}^{3}\cdot {x}^{3}[/latex] |
| Notice that each factor was raised to the power. | [latex]{\left(2x\right)}^{3}\text{ is }{2}^{3}\cdot {x}^{3}[/latex] |
| We write: | [latex]{\left(2x\right)}^{3}[/latex]
[latex]{2}^{3}\cdot {x}^{3}[/latex] |
The exponent applies to each of the factors. This leads to the Product to a Power Property for Exponents.
Product to a Power Property of Exponents
If [latex]a[/latex] and [latex]b[/latex] are real numbers and [latex]m[/latex] is a whole number, then
[latex]{\left(ab\right)}^{m}={a}^{m}{b}^{m}[/latex]
To raise a product to a power, raise each factor to that power.
An example with numbers helps to verify this property:
[latex]\begin{array}{ccc}\hfill {\left(2\cdot 3\right)}^{2}& \stackrel{?}{=}& {2}^{2}\cdot {3}^{2}\hfill \\ \hfill {6}^{2}& \stackrel{?}{=}& 4\cdot 9\hfill \\ \hfill 36& =& 36\hfill \end{array}[/latex]
example
Simplify: [latex]{\left(-11x\right)}^{2}[/latex]
try it
example
Simplify: [latex]{\left(3xy\right)}^{3}[/latex]
try it
In the next video we show more examples of how to simplify a product raised to a power.
