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6.3: Power Rule for Exponents

Difficulty Level: At Grade Created by: CK-12
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Can you simplify an expression where an exponent has an exponent? For example, how would you simplify \begin{align*}[(2^3)^2]^4\end{align*}?

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James Sousa: Properties of Exponents

Guidance

In the expression \begin{align*}x^3\end{align*}, the \begin{align*}x\end{align*} is called the base and the \begin{align*}3\end{align*} is called the exponent. Exponents are often referred to as powers. When an exponent is a positive whole number, it tells you how many times to multiply the base by itself. For example:

  • \begin{align*}x^3=x\cdot x \cdot x\end{align*}
  • \begin{align*}2^4=2\cdot 2 \cdot 2 \cdot 2=16\end{align*}.

There are many rules that have to do with exponents (often called the Laws of Exponents) that are helpful to know so that you can work with expressions and equations that involve exponents more easily. Here you will learn a rule that has to do with raising a power to another power.

RULE: To raise a power to a new power, multiply the exponents.

\begin{align*}& (a^m)^n = \underleftrightarrow{(a \times a \times \ldots \times a)^n}\\ & \qquad \qquad \qquad \quad {\color{red}\downarrow}\\ & \qquad \qquad \quad {\color{red}m} \ \text{{\color{red} factors}}\\ & (a^m)^n=\underleftrightarrow{(a \times a \times \ldots \times a)} \times \underleftrightarrow{(a \times a \times \ldots \times a)} \ \underleftrightarrow{(a \times a \times \ldots \times a)}\\ & \qquad \qquad \qquad \quad \ {\color{red}\downarrow} \qquad \qquad \qquad \qquad {\color{red}\downarrow} \qquad \qquad \qquad \quad {\color{red}\downarrow}\\ & \qquad \quad \quad \underleftrightarrow{\quad {\color{red}m} \ \text{{\color{red}factors}} \qquad \qquad \ \ {\color{red}m} \ \text{{\color{red}factors}} \qquad \quad \ {\color{red}m} \ \text{{\color{red}factors}} \ \ }\\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad {\color{red}n \ times}\\ & (a^m)^n=\underleftrightarrow{a \times a \times a \ldots \times a}\\ & \qquad \qquad {\color{red}mn \ \text{factors}}\\ & (a^m)^n = a^{\color{red}mn}\end{align*}

Example A

Evaluate \begin{align*}(2^3)^2\end{align*}.

Solution:

\begin{align*}& (2^3)^2 && \text{The base is} \ ‘2^3’.\\ & 2^{3 \times 2} && \text{Keep the base of} \ ‘2^3’ \ \text{and multiply the exponents}.\\ & 2^{\color{red}6} && \text{The answer is in exponential form.}\end{align*}

The answer can be taken one step further. The base is numerical so the term can be evaluated.

\begin{align*}& 2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2\\ & {\color{red}2^6}={\color{red}64}\\ & \boxed{(2^3)^2=2^6=64}\end{align*}

Example B

Evaluate \begin{align*}(x^7)^4\end{align*}.

Solution:

\begin{align*}& (x^7)^4 && \text{The base is} \ ‘x^7’.\\ & x^{7 \times 4} && \text{Keep the base of} \ ‘x^7’ \ \text{and multiply the exponents}.\\ & x^{\color{red}28} && \text{The answer is in exponential form.}\\ & \boxed{(x^7)^4 = x^{28}}\end{align*}

Example C

Evaluate \begin{align*}(3^2)^3\end{align*}.

Solution:

\begin{align*}& (3^2)^3 && \text{The base is} \ ‘3^2’.\\ & 3^{2 \times 3} && \text{Keep the base of} \ ‘3^2’ \ \text{and multiply the exponents}.\\ & 3^{\color{red}6} && \text{The answer is in exponential form}.\end{align*}

The answer can be taken one step further. The base is numerical so the term can be evaluated.

\begin{align*}& 3^6=3 \times 3 \times 3 \times 3 \times 3 \times 3\\ & {\color{red}3^6} = {\color{red}729}\\ & \boxed{(3^2)^3=3^6=729}\end{align*}

Example D

Evaluate \begin{align*}(y^4)^2\end{align*}.

Solution:

\begin{align*}& (y^4)^2 && \text{The base is} \ ‘y^4’.\\ & y^{4 \times 2} && \text{Keep the base of} \ ‘y^4’ \ \text{and multiply the exponents}.\\ & y^{\color{red}8} && \text{The answer is in exponential form}.\\ & \boxed{(y^4)^2=y^8}\end{align*}

Concept Problem Revisited

\begin{align*}[(2^3)^2]^4=[2^6]^4=2^24\end{align*}. Notice that the power rule applies even when a number has been raised to more than one powers. The overall exponent is 24 which is \begin{align*}3\cdot 2 \cdot 4\end{align*}.

Vocabulary

Base
In an algebraic expression, the base is the variable, number, product or quotient, to which the exponent refers. Some examples are: In the expression \begin{align*}2^5\end{align*}, ‘2’ is the base. In the expression \begin{align*}(-3y)^4\end{align*}, ‘\begin{align*}-3y\end{align*}’ is the base.
Exponent
In an algebraic expression, the exponent is the number to the upper right of the base that tells how many times to multiply the base times itself. Some examples are:
In the expression \begin{align*}2^5\end{align*}, ‘5’ is the exponent. It means to multiply 2 times itself 5 times as shown here: \begin{align*}2^5=2 \times 2 \times 2 \times 2 \times 2\end{align*}
In the expression \begin{align*}(-3y)^4\end{align*}, ‘4’ is the exponent. It means to multiply \begin{align*}-3y\end{align*} times itself 4 times as shown here: \begin{align*}(-3y)^4=-3y \times -3y \times -3y \times -3y\end{align*}.
Laws of Exponents
The laws of exponents are the algebra rules and formulas that tell us the operation to perform on the exponents when dealing with exponential expressions.

Guided Practice

You know you can rewrite \begin{align*}2^4\end{align*} as \begin{align*}2 \times 2 \times 2 \times 2\end{align*} and then calculate in order to find that \begin{align*}\boxed{2^4=16}\end{align*}. This concept can also be reversed. To write 32 as a power of 2, \begin{align*}32=2 \times 2 \times 2 \times 2 \times 2\end{align*}. There are 5 twos; therefore, \begin{align*}\boxed{32=2^{\color{red}5}}\end{align*}. Use this idea to complete the following exercises.

1. Write 81 as a power of 3.

2. Write \begin{align*}(9)^3\end{align*} as a power of 3.

3. Write \begin{align*}(4^3)^2\end{align*} as a power of 2.

Answers:

1. \begin{align*}81={\color{red}3} \times {\color{red}3}=9 \times {\color{red}3}=27 \times {\color{red}3}=81\end{align*}

There are 4 threes. Therefore \begin{align*}\boxed{81=3^{\color{red}4}}\end{align*}

2. \begin{align*}9={\color{red}3} \times {\color{red}3}=9\end{align*}

There are 2 threes. Therefore \begin{align*}\boxed{9=3^{\color{red}2}}\end{align*}.
\begin{align*}(3^2)^3\end{align*} Apply the law of exponents for power to a power-multiply the exponents.
\begin{align*}3^{2 \times 3}=3^{\color{red}6}\end{align*}
Therefore \begin{align*}\boxed{(9)^3=3^{\color{red}6}}\end{align*}

3. \begin{align*}4={\color{red}2} \times {\color{red}2}=4\end{align*}

There are 2 twos. Therefore \begin{align*}\boxed{4=2^{\color{red}2}}\end{align*}
\begin{align*}\left((2^2)^3\right)^2\end{align*} Apply the law of exponents for power to a power-multiply the exponents.
\begin{align*}\boxed{2^{2 \times 3}=2^{\color{red}6}}\end{align*}
\begin{align*}(2^6)^2\end{align*} Apply the law of exponents for power to a power-multiply the exponents.
\begin{align*}\boxed{2^{6 \times 2}=2^{\color{red}12}}\end{align*}
Therefore \begin{align*}\boxed{(4^3)^2=2^{\color{red}12}}\end{align*}

Practice

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Difficulty Level:
At Grade
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Date Created:
Jan 16, 2013
Last Modified:
Apr 29, 2014

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