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6.6: Exponential Expressions

Difficulty Level: Advanced Created by: CK-12
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Practice Evaluating Exponential Expressions
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Can you simplify the following expression so that it has only positive exponents?

\frac{8x^3y^{-2}}{(-4a^2b^4)^{-2}}

Watch This

James Sousa: Simplify Exponential Expressions

Guidance

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:

\left(\frac{1}{81}\right)^{\frac{1}{4}}=\sqrt[4]{\frac{1}{81}}=\frac{1}{3}

Example B

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

Solution:

(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} .

Solution:

\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

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}

Answers:

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}}

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Difficulty Level:

Advanced

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Date Created:

Apr 30, 2013

Last Modified:

Feb 26, 2015
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