<meta http-equiv="refresh" content="1; url=/nojavascript/"> Exponential Expressions | CK-12 Foundation
Skip Navigation

6.6: Exponential Expressions

Difficulty Level: Advanced Created by: CK-12
Practice Evaluating Exponential Expressions
Practice Now

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


Watch This

James Sousa: Simplify Exponential Expressions


The following table summarizes all of the rules for exponents.

Laws of Exponents

If a \in R, a \ge 0 and m, n \in Q , then

  1. a^m \times a^n=a^{m+n}
  2. \frac{a^m}{a^n}=a^{m-n} \ (\text{if} \ m > n, a \neq 0)
  3. (a^m)^n=a^{mn}
  4. (ab)^n=a^nb^n
  5. \left(\frac{a}{b}\right)^n=\frac{a^n}{b^n} \ (b \neq 0)
  6. a^0=1 \ (a \neq 0)
  7. a^{-m}=\frac{1}{a^m}
  8. a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m

Example A

Evaluate 81^{-\frac{1}{4}} .

Solution: First, rewrite with a positive exponent:

81^{-\frac{1}{4}}=\frac{1}{81^{\frac{1}{4}}}=\left(\frac{1}{81}\right)^{\frac{1}{4}} .

Next, evaluate the fractional exponent:


Example B

Simplify (4x^3 y) (3x^5 y^2 )^4  .


(4x^3 y) (3x^5 y^2 )^4&=(4x^3 y) (81x^{20} y^8 )\\ & =324x^{23}y^9

Example C

Simplify \left(\frac{x^{-2}y}{x^4y^3}\right)^{-2} .


\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

Concept Problem Revisited

\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


In an algebraic expression, the base is the variable, number, product or quotient, to which the exponent refers. Some examples are: In the expression 2^5 , ‘2’ is the base. In the expression (-3y)^4 , ‘ -3y ’ is the base.
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 2^5 , ‘5’ is the exponent. It means to multiply 2 times itself 5 times as shown here: 2^5=2 \times 2 \times 2 \times 2 \times 2 .
In the expression (-3y)^4 , ‘4’ is the exponent. It means to multiply -3y times itself 4 times as shown here: (-3y)^4=-3y \times -3y \times -3y \times -3y .
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

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

1. (-2x)^5 (2x^2)

2. (16x^{10}) \left(\frac{3}{4}x^5\right)

3. \frac{(x^{15})(x^{24})(x^{25})}{(x^7)^8}


1. (-2x)^5 (2x^2)=(-32x^5)(2x^2)=-64x^7

2. (16x^{10}) \left(\frac{3}{4}x^5\right)=12x^{15}

3. \frac{(x^{15})(x^{24})(x^{25})}{(x^7)^8}=\frac{x^{64}}{x^{56}}=x^8

Explore More

Simplify each expression.

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

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

  1. (3^3)^5
  2. (3^9)(3^3)
  3. (9)(3^7)
  4. 9^4
  5. (9)(27^2)

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

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

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

  1. (4x)^2
  2. (-3x)^3
  3. (x^3)^4
  4. (3x)(x^7)
  5. (5x)(4x^4)
  6. (-3x^2)(-6x^3)
  7. (10x^8) \div (2x^4)

Simplify each of the following using the laws of exponents.

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

Image Attributions


Difficulty Level:



Date Created:

Apr 30, 2013

Last Modified:

Dec 23, 2014
You can only attach files to Modality which belong to you
If you would like to associate files with this Modality, please make a copy first.


Please wait...
Please wait...
Image Detail
Sizes: Medium | Original

Original text