<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Dismiss
Skip Navigation
Our Terms of Use (click here to view) have changed. By continuing to use this site, you are agreeing to our new Terms of Use.

8.7: Evaluating Exponential Expressions

Difficulty Level: At Grade Created by: CK-12
Atoms Practice
Estimated11 minsto complete
%
Progress
Practice Evaluating Exponential Expressions
Practice
Progress
Estimated11 minsto complete
%
Practice Now
Turn In

What if you had an exponential expression requiring multiple operations, like \begin{align*}2\left(\frac{1}{4}\right)^2 - \left(\frac{1}{4}\right)^3\end{align*}? How could you simplify it? After completing this Concept, you'll be able to use the order of operations to evaluate exponential expressions like this one.

Watch This

Foundation: 02807S Evaluating Exponential Expressions

Guidance

When evaluating expressions we must keep in mind the order of operations. You must remember PEMDAS:

  1. Evaluate inside the Parentheses.
  2. Evaluate Exponents.
  3. Perform Multiplication and Division operations from left to right.
  4. Perform Addition and Subtraction operations from left to right.

Example A

Evaluate the following expressions.

a) \begin{align*}5^0\end{align*}

b) \begin{align*}\left(\frac{2}{3}\right)^3\end{align*}

c) \begin{align*}16^{\frac{1}{2}}\end{align*}

d) \begin{align*}8^{-\frac{1}{3}}\end{align*}

Solution

a) \begin{align*}5^0=1\end{align*} A number raised to the power 0 is always 1.

b) \begin{align*}\left(\frac{2}{3}\right)^3=\frac{2^3}{3^3}=\frac{8}{27}\end{align*}

c) \begin{align*}16^{\frac{1}{2}}=\sqrt{16}=4\end{align*} Remember that an exponent of \begin{align*}\frac{1}{2}\end{align*} means taking the square root.

d) \begin{align*}8^{-\frac{1}{3}}=\frac{1}{8^{\frac{1}{3}}}=\frac{1}{\sqrt[3]{8}}=\frac{1}{2}\end{align*} Remember that an exponent of \begin{align*}\frac{1}{3}\end{align*} means taking the cube root.

Example B

Evaluate the following expressions.

a) \begin{align*}3 \cdot 5^2-10 \cdot 5+1\end{align*}

b) \begin{align*}\frac{2 \cdot 4^2-3 \cdot 5^2}{3^2-2^2}\end{align*}

c) \begin{align*}\left(\frac{3^3}{2^2}\right)^{-2} \cdot \frac{3}{4}\end{align*}

Solution

a) Evaluate the exponent: \begin{align*}3 \cdot 5^2 - 10 \cdot 5+1=3 \cdot 25-10 \cdot 5+1\end{align*}

Perform multiplications from left to right: \begin{align*}3 \cdot 25-10 \cdot 5+1=75-50+1\end{align*}

Perform additions and subtractions from left to right: \begin{align*}75-50+1=26\end{align*}

b) Treat the expressions in the numerator and denominator of the fraction like they are in parentheses: \begin{align*}\frac{(2 \cdot 4^2-3 \cdot 5^2)}{(3^2-2^2)}=\frac{(2 \cdot 16-3 \cdot 25)}{(9-4)}=\frac{(32-75)}{5}=\frac{-43}{5}\end{align*}

c) \begin{align*}\left(\frac{3^3}{2^2}\right)^{-2} \cdot \frac{3}{4}=\left(\frac{2^2}{3^3}\right)^2 \cdot \frac{3}{4}=\frac{2^4}{3^6} \cdot \frac{3}{4}=\frac{2^4}{3^6} \cdot \frac{3}{2^2}=\frac{2^2}{3^5}=\frac{4}{243}\end{align*}

Example C

Evaluate the following expressions for \begin{align*}x = 2, y = - 1, z = 3\end{align*}.

a) \begin{align*}2x^2-3y^3+4z\end{align*}

b) \begin{align*}(x^2-y^2)^2\end{align*}

c) \begin{align*}\left(\frac{3x^2y^5}{4z}\right)^{-2}\end{align*}

Solution

a) \begin{align*}2x^2-3y^3+4z&=2 \cdot 2^2-3 \cdot (-1)^3+4 \cdot 3\\ &=2 \cdot 4-3 \cdot (-1)+4 \cdot 3=8+3+12\\ &=23\end{align*}

b) \begin{align*}(x^2-y^2)^2=(2^2 - (-1)^2)^2=(4-1)^2=3^2=9\end{align*}

c)

\begin{align*}\left( \frac{3x^2y^5}{4z}\right)^{-2}&=\left( \frac{3 \cdot 2^2 \cdot (-1)^5}{4 \cdot 3}\right)^{-2}\\ &=\left(\frac{3 \cdot 4 \cdot (-1)}{12}\right)^{-2}\\ &=\left(\frac{-12}{12}\right)^{-2}\\ &=\left(\frac{-1}{1}\right)^{-2}\\ &=\left(\frac{1}{-1}\right)^2\\ &=(-1)^2\\ &=1\end{align*}

Watch this video for help with the Examples above.

CK-12 Foundation: Evaluating Exponential Expressions

Vocabulary

  • When evaluating expressions we must keep in mind the order of operations. You must remember PEMDAS:
  1. Evaluate inside the Parentheses.
  2. Evaluate Exponents.
  3. Perform Multiplication and Division operations from left to right.
  4. Perform Addition and Subtraction operations from left to right.

Guided Practice

Evaluate the following expression for \begin{align*}x = 3, y = -2, z = -1\end{align*}.

\begin{align*}2z((x+1)^\frac{1}{2}-y^3)^2\end{align*}

Solution:

\begin{align*}2z((x+1)^\frac{1}{2}-y^3)^2&=2(-1)(((3)+1)^\frac{1}{2}-(-2)^3)^2\\ &=-2(4^\frac{1}{2}+8)^2\\ &=-2(2+8)^2\\ &=-2(10)^2\\ &=-200\end{align*}

Practice

Evaluate the following expressions to a single number.

  1. \begin{align*}3^{-2}\end{align*}
  2. \begin{align*}{-4}^{-3}\end{align*}
  3. \begin{align*}(6.2)^0\end{align*}
  4. \begin{align*}8^{-4} \cdot 8^6\end{align*}
  5. \begin{align*}\left (16^\frac{1}{2} \right )^3\end{align*}
  6. \begin{align*}x^2 \cdot 4x^3 \cdot y^4 \cdot 4y^2\end{align*}, if \begin{align*}x = 2\end{align*} and \begin{align*}y = -1\end{align*}
  7. \begin{align*}a^4(b^2)^3 + 2ab\end{align*}, if \begin{align*}a = -2\end{align*} and \begin{align*}b = 1\end{align*}
  8. \begin{align*}5x^2 - 2y^3 + 3z\end{align*}, if \begin{align*}x = 3, y = 2,\end{align*} and \begin{align*}z = 4\end{align*}
  9. \begin{align*}\left ( \frac{a^2}{b^3} \right )^{-2}\end{align*}, if \begin{align*}a = 5\end{align*} and \begin{align*}b = 3\end{align*}
  10. \begin{align*}\left ( \frac{x^{-2}}{y^4} \right )^\frac{1}{2}\end{align*}, if \begin{align*}x=-3\end{align*} and \begin{align*}y=2\end{align*}

Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes
Show More

Image Attributions

Show Hide Details
Description
Difficulty Level:
At Grade
Grades:
Date Created:
Oct 01, 2012
Last Modified:
Apr 11, 2016
Files can only be attached to the latest version of Modality
Please wait...
Please wait...
Image Detail
Sizes: Medium | Original
 
MAT.ALG.722.3.L.1
Here