<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Dismiss
Skip Navigation

6.6: Exponential Expressions

Difficulty Level: Advanced Created by: CK-12
Atoms Practice
Estimated12 minsto complete
%
Progress
Practice Evaluating Exponential Expressions
Practice
Progress
Estimated12 minsto complete
%
Practice Now
Turn In

Can you simplify the following expression so that it has only positive exponents? 

\begin{align*}\frac{8x^3y^{-2}}{(-4a^2b^4)^{-2}}\end{align*}8x3y2(4a2b4)2

Exponential Expressions

Here is a summary of all the Laws of Exponents that have been covered so far:

Laws of Exponents

If \begin{align*}a \in R, a \ge 0\end{align*}aR,a0 and \begin{align*}m, n \in Q\end{align*}m,nQ, then

  1. \begin{align*}a^m \times a^n=a^{m+n}\end{align*}am×an=am+n
  2. \begin{align*}\frac{a^m}{a^n}=a^{m-n} \ (\text{if} \ m > n, a \neq 0)\end{align*}aman=amn (if m>n,a0)
  3. \begin{align*}(a^m)^n=a^{mn}\end{align*}(am)n=amn
  4. \begin{align*}(ab)^n=a^nb^n\end{align*}(ab)n=anbn
  5. \begin{align*}\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n} \ (b \neq 0)\end{align*}(ab)n=anbn (b0)
  6. \begin{align*}a^0=1 \ (a \neq 0)\end{align*}a0=1 (a0)
  7. \begin{align*}a^{-m}=\frac{1}{a^m}\end{align*}am=1am
  8. \begin{align*}a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m\end{align*}amn=amn=(an)m

Now, applying the Laws of Exponents, let's evaluate and simplify the following expressions:

  1. \begin{align*}81^{-\frac{1}{4}}\end{align*}8114

First, rewrite with a positive exponent:

\begin{align*}81^{-\frac{1}{4}}=\frac{1}{81^{\frac{1}{4}}}=\left(\frac{1}{81}\right)^{\frac{1}{4}}\end{align*}8114=18114=(181)14.

Next, evaluate the fractional exponent:

\begin{align*}\left(\frac{1}{81}\right)^{\frac{1}{4}}=\sqrt[4]{\frac{1}{81}}=\frac{1}{3}\end{align*}(181)14=1814=13

  1. \begin{align*}(4x^3 y) (3x^5 y^2 )^4 \end{align*}(4x3y)(3x5y2)4

\begin{align*}(4x^3 y) (3x^5 y^2 )^4&=(4x^3 y) (81x^{20} y^8 )\\ & =324x^{23}y^9\end{align*}(4x3y)(3x5y2)4=(4x3y)(81x20y8)=324x23y9

  1. \begin{align*}\left(\frac{x^{-2}y}{x^4y^3}\right)^{-2}\end{align*}(x2yx4y3)2

\begin{align*}\left(\frac{x^{-2}y}{x^4y^3}\right)^{-2}&=\left(\frac{x^4y^3}{x^{-2}y}\right)^{2}\\ &=(x^6y^2)^{2}\\ &=x^{12}y^4\end{align*}(x2yx4y3)2=(x4y3x2y)2=(x6y2)2=x12y4

Examples

Example 1

Earlier, you were asked to simplify the following expression:

\begin{align*}\frac{8x^3y^{-2}}{(-4a^2b^4)^{-2}}\end{align*}8x3y2(4a2b4)2

\begin{align*}\frac{8x^3y^{-2}}{(-4x^2y^4)^{-2}}&=(8x^3y^{-2})(-4x^2y^4)^2\\ &=(8x^3y^{-2})(16x^4y^8) \\ &=8\cdot 16 \cdot x^3 \cdot x^4 \cdot y^{-2} \cdot y^8\\ &=128x^7y^6\end{align*}8x3y2(4x2y4)2=(8x3y2)(4x2y4)2=(8x3y2)(16x4y8)=816x3x4y2y8=128x7y6

Example 2

Use the Laws of Exponents to simplify the following:

 \begin{align*}(-2x)^5 (2x^2)\end{align*}(2x)5(2x2)

 \begin{align*}(-2x)^5 (2x^2)=(-32x^5)(2x^2)=-64x^7\end{align*}(2x)5(2x2)=(32x5)(2x2)=64x7

Example 3

Use the Laws of Exponents to simplify the following: 

\begin{align*}(16x^{10}) \left(\frac{3}{4}x^5\right)\end{align*}(16x10)(34x5)

\begin{align*}(16x^{10}) \left(\frac{3}{4}x^5\right)=12x^{15}\end{align*}(16x10)(34x5)=12x15

Example 4

Use the Laws of Exponents to simplify the following: 

 \begin{align*}\frac{(x^{15})(x^{24})(x^{25})}{(x^7)^8}\end{align*}(x15)(x24)(x25)(x7)8

\begin{align*}\frac{(x^{15})(x^{24})(x^{25})}{(x^7)^8}=\frac{x^{64}}{x^{56}}=x^8\end{align*}(x15)(x24)(x25)(x7)8=x64x56=x8

Review

Simplify each expression.

  1. \begin{align*}(x^{10}) (x^{10})\end{align*}
  2. \begin{align*}(7x^3)(3x^7)\end{align*}
  3. \begin{align*}(x^3 y^2) (xy^3) (x^5 y)\end{align*}
  4. \begin{align*}\frac{(x^3)(x^2)}{(x^4)}\end{align*}
  5. \begin{align*}\frac{x^2}{x^{-3}}\end{align*}
  6. \begin{align*}\frac{x^6 y^8}{x^4 y^{-2}}\end{align*}
  7. \begin{align*}(2x^{12})^3\end{align*}
  8. \begin{align*}(x^5 y^{10})^7\end{align*}
  9. \begin{align*}\left(\frac{2x^{10}}{3y^{20}}\right)^3\end{align*}

Express each of the following as a power of 3. Do not evaluate.

  1. \begin{align*}(3^3)^5\end{align*}
  2. \begin{align*}(3^9)(3^3)\end{align*}
  3. \begin{align*}(9)(3^7)\end{align*}
  4. \begin{align*}9^4\end{align*}
  5. \begin{align*}(9)(27^2)\end{align*}

Apply the laws of exponents to evaluate each of the following without using a calculator.

  1. \begin{align*}(2^3)(2^2)\end{align*}
  2. \begin{align*}6^6 \div 6^5\end{align*}
  3. \begin{align*}-(3^2)^3\end{align*}
  4. \begin{align*}(1^2)^3+(1^3)^2\end{align*}
  5. \begin{align*}\left(\frac{1}{3}\right)^6 \div \left(\frac{1}{3}\right)^8\end{align*}

Use the laws of exponents to simplify each of the following.

  1. \begin{align*}(4x)^2\end{align*}
  2. \begin{align*}(-3x)^3\end{align*}
  3. \begin{align*}(x^3)^4\end{align*}
  4. \begin{align*}(3x)(x^7)\end{align*}
  5. \begin{align*}(5x)(4x^4)\end{align*}
  6. \begin{align*}(-3x^2)(-6x^3)\end{align*}
  7. \begin{align*}(10x^8) \div (2x^4)\end{align*}

Simplify each of the following using the laws of exponents.

  1. \begin{align*}5^{\frac{1}{2}} \times 5^{\frac{1}{3}}\end{align*}
  2. \begin{align*}(d^4 e^8 f^{12})^{\frac{1}{4}}\end{align*}
  3. \begin{align*}\sqrt[4]{\frac{y^{\frac{1}{2}} \sqrt{xy}}{x^{\frac{2}{3}}}}\end{align*}
  4. \begin{align*}(32a^{20}b^{-15})^{\frac{1}{5}}\end{align*}
  5. \begin{align*}(729x^{12}y^{-6})^{\frac{2}{3}}\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 6.6. 

Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes
Show More

Image Attributions

Show Hide Details
Description
Difficulty Level:
Advanced
Grades:
Date Created:
Apr 30, 2013
Last Modified:
Mar 23, 2016
Save or share your relevant files like activites, homework and worksheet.
To add resources, you must be the owner of the Modality. Click Customize to make your own copy.
Please wait...
Please wait...
Image Detail
Sizes: Medium | Original
 
MAT.ALG.722.3.L.2
Here