Simplifying Variable Expressions Using Exponent Properties

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.

	${x}^{2}\cdot{x}^{3}$
What does this mean? How many factors altogether?
So, we have	${x}{5}$
Notice that $5$ is the sum of the exponents, $2$ and $3$ .	${x}^{2}\cdot{x}^{3}$ is ${x}^{2+3}$ , or ${x}^{5}$
We write:	${x}^{2}\cdot {x}^{3}$ ${x}^{2+3}$ ${x}^{5}$

The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.

Product Property of Exponents

a

is a real number and

m,n

are counting numbers, then

{a}^{m}\cdot {a}^{n}={a}^{m+n}

To multiply with like bases, add the exponents.

An example with numbers helps to verify this property.

\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}

example

Simplify:

{x}^{5}\cdot {x}^{7}

. Solution

	${x}^{5}\cdot {x}^{7}$
Use the product property, ${a}^{m}\cdot {a}^{n}={a}^{m+n}$ .	$x^{\color{red}{5+7}}$
Simplify.	${x}^{12}$

try it

[ohm_question]146102[/ohm_question]

example

Simplify:

{b}^{4}\cdot b

Answer: Solution

	${b}^{4}\cdot b$
Rewrite, $b={b}^{1}$ .	${b}^{4}\cdot {b}^{1}$
Use the product property, ${a}^{m}\cdot {a}^{n}={a}^{m+n}$ .	$b^{\color{red}{4+1}}$
Simplify.	${b}^{5}$

try it

[ohm_question]146107[/ohm_question]

example

Simplify:

{2}^{7}\cdot {2}^{9}

Answer: Solution

	${2}^{7}\cdot {2}^{9}$
Use the product property, ${a}^{m}\cdot {a}^{n}={a}^{m+n}$ .	$2^{\color{red}{7+9}}$
Simplify.	${2}^{16}$

try it

[ohm_question]146143[/ohm_question]

example

Simplify:

{y}^{17}\cdot {y}^{23}

Answer: Solution

	${y}^{17}\cdot {y}^{23}$
Notice, the bases are the same, so add the exponents.	$y^{\color{red}{17+23}}$
Simplify.	${y}^{40}$

try it

[ohm_question]146144[/ohm_question]

We can extend the Product Property of Exponents to more than two factors.

example

Simplify:

{x}^{3}\cdot {x}^{4}\cdot {x}^{2}

Answer: Solution

	${x}^{3}\cdot {x}^{4}\cdot {x}^{2}$
Add the exponents, since the bases are the same.	$x^{\color{red}{3+4+2}}$
Simplify.	${x}^{9}$

try it

[ohm_question]146145[/ohm_question]

In the following video we show more examples of how to use the product rule for exponents to simplify expressions. https://youtu.be/P0UVIMy2nuI

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.

	$({x}^{2})^{3}$
	${x}^{2}\cdot{x}^{2}\cdot{x}^{2}$
What does this mean? How many factors altogether?
So, we have	${x}^{6}$
Notice that $6$ is the product of the exponents, $2$ and $3$ .	$({x}^{2})^{3}$ is ${x}^{2\cdot3}$ or ${x}^{6}$
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.

Power Property of Exponents

a

is a real number and

m,n

are whole numbers, then

{\left({a}^{m}\right)}^{n}={a}^{m\cdot 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\cdot 3}\hfill \\ \hfill {\left(25\right)}^{3}& \stackrel{?}{=}& {5}^{6}\hfill \\ \hfill 15,625& =& 15,625\hfill \end{array}

example

Simplify: 1.

{\left({x}^{5}\right)}^{7}

{\left({3}^{6}\right)}^{8}

Answer: Solution

1.
	${\left({x}^{5}\right)}^{7}$
Use the Power Property, ${\left({a}^{m}\right)}^{n}={a}^{m\cdot n}$ .	$x^{\color{red}{5\cdot{7}}}$
Simplify.	${x}^{35}$

2.
	${\left({3}^{6}\right)}^{8}$
Use the Power Property, ${\left({a}^{m}\right)}^{n}={a}^{m\cdot n}$ .	$3^{\color{red}{6\cdot{8}}}$
Simplify.	${3}^{48}$

try it

[ohm_question]146148[/ohm_question]

Watch the following video to see more examples of how to use the power rule for exponents to simplify expressions. https://youtu.be/Hgu9HKDHTUA

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.

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

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

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\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}

example

Simplify:

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

Answer: Solution

	${\left(-11x\right)}^{2}$
Use the Power of a Product Property, ${\left(ab\right)}^{m}={a}^{m}{b}^{m}$ .	$(-11)^{\color{red}{2}}x^{\color{red}{2}}$
Simplify.	$121{x}^{2}$

try it

[ohm_question]146152[/ohm_question]

example

Simplify:

{\left(3xy\right)}^{3}

Answer: Solution

	${\left(3xy\right)}^{3}$
Raise each factor to the third power.	$3^{\color{red}{3}}x^{\color{red}{3}}y^{\color{red}{3}}$
Simplify.	$27{x}^{3}{y}^{3}$

try it

[ohm_question]146154[/ohm_question]

In the next video we show more examples of how to simplify a product raised to a power. https://youtu.be/D05D-YIPr1Q

Simplifying Variable Expressions Using Exponent Properties

Learning Outcomes

Product Property of Exponents

example

try it

example

try it

example

try it

example

try it

example

try it

Simplify Expressions Using the Power Property of Exponents

Power Property of Exponents

example

try it

Simplify Expressions Using the Product to a Power Property

Product to a Power Property of Exponents

example

try it

example

try it

Licenses & Attributions

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CC licensed content, Shared previously

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Guides d'étude > Prealgebra

Simplifying Variable Expressions Using Exponent Properties

Learning Outcomes

Product Property of Exponents

example

try it

example

try it

example

try it

example

try it

example

try it

Simplify Expressions Using the Power Property of Exponents

Power Property of Exponents

example

try it

Simplify Expressions Using the Product to a Power Property

Product to a Power Property of Exponents

example

try it

example

try it

Licenses & Attributions

CC licensed content, Original

CC licensed content, Shared previously

CC licensed content, Specific attribution