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6.6: Exponential Expressions

Difficulty Level: At Grade Created by: CK-12

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

Watch This

James Sousa: Simplify Exponential Expressions

Guidance

The following table summarizes all of the rules for exponents.

Laws of Exponents

If \begin{align*}a \varepsilon R, a \ge 0\end{align*}aεR,a0 and \begin{align*}m, n \varepsilon Q\end{align*}m,nεQ, 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

Example A

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

Solution:

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

Example B

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

Solution:

\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

Example C

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

Solution:

\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*}

Concept Problem Revisited

\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*}

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

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

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

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

3. \begin{align*}\frac{(x^{15})(x^{24})(x^{25})}{(x^7)^8}\end{align*}

Answers:

1. \begin{align*}(-2x)^5 (2x^2)=(-32x^5)(2x^2)=-64x^7\end{align*}

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

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

Practice

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*}

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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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