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8.3: Negative Exponents

Difficulty Level: Basic Created by: CK-12
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Practice Negative Exponents

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All the students in a class were randomly given an expression, and they were asked to make pairs, with one boy and one girl per pair. The students were asked to divide the boy's expression by the girl's expression. Bill and Jenna paired up, with Bill having 10x4\begin{align*}10x^4\end{align*} and Jenna having 5x5\begin{align*}5x^5\end{align*}. Also, Tim and Meg paired up, with Tim having 7y3\begin{align*}7y^3\end{align*}, and Meg having 14y3\begin{align*}14y^3\end{align*}. What is Bill and Jenna's quotient? How about Tim and Meg's quotient? In this Concept, you'll learn about zero and negative exponents so that you can perform division problems like these.

Watch This

Multimedia Link: For help with these types of exponents, watch this http://www.phschool.com/atschool/academy123/english/academy123_content/wl-book-demo/ph-241s.html - PH School video or visit the http://www.mathsisfun.com/algebra/negative-exponents.html - mathisfun website.

Guidance

In the previous Concepts, we have dealt with powers that are positive whole numbers. In this Concept, you will learn how to solve expressions when the exponent is zero or a negative number.

Exponents of Zero: For all real numbers χ,χ0,χ0=1\begin{align*}\chi, \chi \neq 0, \chi^0=1\end{align*}.

Example A

Simplify χ4χ4\begin{align*}\frac{\chi^4}{\chi^4}\end{align*}.

Solution:

χ4χ4=χ44=χ0=1\begin{align*}\frac{\chi^4}{\chi^4} = \chi^{4-4} = \chi^0 = 1\end{align*}. This example is simplified using the Quotient of Powers Property.

Simplifying Expressions with Negative Exponents

The next objective is negative exponents. When we use the quotient rule and we subtract a greater number from a smaller number, the answer will become negative. The variable and the power will be moved to the denominator of a fraction. You will learn how to write this in an expression.

Example B

Simplify x4x6\begin{align*}\frac{x^4}{x^6}\end{align*}.

Solution:

x4x6=x46=x2=1x2\begin{align*}\frac{x^4}{x^6} =x^{4-6}=x^{-2}=\frac{1}{x^2}\end{align*}. Another way to look at this is χχχχχχχχχχ\begin{align*}\frac{\chi \cdot \chi \cdot \chi \cdot \chi}{\chi \cdot \chi \cdot \chi \cdot \chi \cdot \chi \cdot \chi}\end{align*}. The four χ\begin{align*}\chi\end{align*}s on top will cancel out with four χ\begin{align*}\chi\end{align*}s on the bottom. This will leave two χ\begin{align*}\chi\end{align*}s remaining on the bottom, which makes your answer look like 1χ2\begin{align*}\frac{1}{\chi^2}\end{align*}.

Negative Power Rule for Exponents: 1χn=χn\begin{align*}\frac{1}{\chi^n} = \chi^{-n}\end{align*} where χ0\begin{align*}\chi \neq 0\end{align*}.

Example C

Rewrite using only positive exponents: χ6γ2\begin{align*}\chi^{-6} \gamma^{-2}\end{align*}.

Solution:

χ6γ2=1χ61γ2=1χ6γ2\begin{align*}\chi^{-6} \gamma^{-2}= \frac{1}{\chi^6} \cdot \frac{1}{\gamma^2} = \frac{1}{\chi^6 \gamma^2}\end{align*}. The negative power rule for exponents is applied to both variables separately in this example.

Example D

Write the following expressions without fractions.

(a) 2x2\begin{align*}\frac{2}{x^2}\end{align*}

(b) x2y3\begin{align*}\frac{x^2}{y^3}\end{align*}

Solution:

(a) 2x2=2x2\begin{align*}\frac{2}{x^2}=2x^{-2}\end{align*}

(b) x2y3=x2y3\begin{align*}\frac{x^2}{y^3}=x^2y^{-3}\end{align*}

Notice in part (a), the number 2 is in the numerator. This number is multiplied with x2\begin{align*}x^{-2}\end{align*}. It could also look like 21x2\begin{align*}2 \cdot \frac{1}{x^2}\end{align*} to be better understood.

Vocabulary

Exponents of zero: For all real numbers χ,χ0,χ0=1\begin{align*}\chi, \chi \neq 0, \chi^0=1\end{align*}.

Negative Power Rule for Exponents: 1χn=χn\begin{align*}\frac{1}{\chi^n} = \chi^{-n}\end{align*} where χ0\begin{align*}\chi \neq 0\end{align*}.

Guided Practice

Simplify (x2y3x5y2)2\begin{align*}\left( \frac{x^2y^{-3}}{x^5y^2}\right)^{2}\end{align*}, giving the answer with only positive exponents.

Solution:

(x2y3x5y2)2=(x2x5y3y2)2=(x25y32)2=(x3y5)2=(x3)2(y5)2=x(3)(2)y(5)(2)=x6y10=1x6y10\begin{align*}\left( \frac{x^2 y^{-3}}{x^5 y^2} \right)^2 = \left(x^2 x^{-5} y^{-3} y^{-2}\right)^2=\left(x^{2-5} y^{-3-2}\right)^2=(x^{-3} y^{-5})^2= (x^{-3})^2 (y^{-5})^2 &= x^{(-3)(2)} y^{(-5)(2)} = x^{-6} y^{-10}=\frac{1}{x^6y^{10}} \end{align*}

Practice

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Zero, Negative, and Fractional Exponents (14:04)

Simplify the following expressions. Be sure the final answer includes only positive exponents.

1. x1y2\begin{align*}x^{-1} \cdot y^2\end{align*}
2. x4\begin{align*}x^{-4}\end{align*}
3. x3x7\begin{align*}\frac{x^{-3}}{x^{-7}}\end{align*}
4. 1x\begin{align*}\frac{1}{x}\end{align*}
5. 2x2\begin{align*}\frac{2}{x^2}\end{align*}
6. x2y3\begin{align*}\frac{x^2}{y^3}\end{align*}
7. 3xy\begin{align*}\frac{3}{xy}\end{align*}
8. 3x3\begin{align*}3x^{-3}\end{align*}
9. a2b3c1\begin{align*}a^2b^{-3}c^{-1}\end{align*}
10. 4x1y3\begin{align*}4x^{-1}y^3\end{align*}
11. 2x2y3\begin{align*}\frac{2x^{-2}}{y^{-3}}\end{align*}
12. \begin{align*}\left(\frac{a}{b}\right)^{-2}\end{align*}
13. \begin{align*}(3a^{-2}b^2c^3)^3\end{align*}
14. \begin{align*}x^{-3} \cdot x^3\end{align*}

Simplify the following expressions without any fractions in the answer.

1. \begin{align*}\frac{a^{-3}(a^5)}{a^{-6}}\end{align*}
2. \begin{align*}\frac{5x^6y^2}{x^8y}\end{align*}
3. \begin{align*}\frac{(4ab^6)^3}{(ab)^5}\end{align*}

Evaluate the following expressions to a single number.

1. \begin{align*}3^{-2}\end{align*}
2. \begin{align*}(6.2)^0\end{align*}
3. \begin{align*}8^{-4} \cdot 8^6\end{align*}

In 21 – 23, evaluate the expression for \begin{align*}x=2, y=-1, \text{and } z=3\end{align*}.

1. \begin{align*}2x^2-3y^3+4z\end{align*}
2. \begin{align*}(x^2-y^2)^2\end{align*}
3. \begin{align*}\left(\frac{3x^2y^5}{4z}\right)^{-2}\end{align*}
4. Evaluate \begin{align*}x^24x^3y^44y^2\end{align*} if \begin{align*}x=2\end{align*} and \begin{align*}y=-1\end{align*}.
5. Evaluate \begin{align*}a^4(b^2)^3+2ab\end{align*} if \begin{align*}a=-2\end{align*} and \begin{align*}b=1\end{align*}.
6. Evaluate \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*}.
7. Evaluate \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*}.
8. Evaluate \begin{align*}3 \cdot 5^5 - 10 \cdot 5+1\end{align*}.
9. Evaluate \begin{align*}\frac{2 \cdot 4^2-3 \cdot 5^2}{3^2}\end{align*}.
10. Evaluate \begin{align*}\left(\frac{3^3}{2^2}\right)^{-2} \cdot \frac{3}{4}\end{align*}.

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Vocabulary Language: English Spanish

TermDefinition
exponents of zero For all real numbers $\chi, \chi \neq 0, \chi^0=1$.
Negative Power Rule for Exponents $\frac{1}{\chi^n} = \chi^{-n}$ where $\chi \neq 0$.
Negative Exponent Property The negative exponent property states that $\frac{1}{a^m} = a^{-m}$ and $\frac{1}{a^{-m}} = a^m$ for $a \neq 0$.
quotient rule In calculus, the quotient rule states that if $f$ and $g$ are differentiable functions at $x$ and $g(x) \ne 0$, then $\frac {d}{dx}\left [ \frac{f(x)}{g(x)} \right ]= \frac {g(x) \frac {d}{dx}\left [{f(x)} \right ] - f(x) \frac{d}{dx} \left [{g(x)} \right ]}{\left [{g(x)} \right ]^2}$.
Zero Exponent Property The zero exponent property says that for all $a \neq 0$, $a^0 = 1$.

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