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# Negative Exponents

## Any value to the zero power equals 1, convert negative exponents

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Negative Exponents

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 \begin{align*}10x^4\end{align*} and Jenna having \begin{align*}5x^5\end{align*}. Also, Tim and Meg paired up, with Tim having \begin{align*}7y^3\end{align*}, and Meg having \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 \begin{align*}\chi, \chi \neq 0, \chi^0=1\end{align*}.

#### Example A

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

Solution:

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

Solution:

\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 \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 \begin{align*}\chi\end{align*}s on top will cancel out with four \begin{align*}\chi\end{align*}s on the bottom. This will leave two \begin{align*}\chi\end{align*}s remaining on the bottom, which makes your answer look like \begin{align*}\frac{1}{\chi^2}\end{align*}.

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

#### Example C

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

Solution:

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

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

Solution:

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

(b) \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 \begin{align*}x^{-2}\end{align*}. It could also look like \begin{align*}2 \cdot \frac{1}{x^2}\end{align*} to be better understood.

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### Guided Practice

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

Solution:

### Explore More

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. \begin{align*}x^{-1} \cdot y^2\end{align*}
2. \begin{align*}x^{-4}\end{align*}
3. \begin{align*}\frac{x^{-3}}{x^{-7}}\end{align*}
4. \begin{align*}\frac{1}{x}\end{align*}
5. \begin{align*}\frac{2}{x^2}\end{align*}
6. \begin{align*}\frac{x^2}{y^3}\end{align*}
7. \begin{align*}\frac{3}{xy}\end{align*}
8. \begin{align*}3x^{-3}\end{align*}
9. \begin{align*}a^2b^{-3}c^{-1}\end{align*}
10. \begin{align*}4x^{-1}y^3\end{align*}
11. \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*}.

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 8.3.

### Vocabulary Language: English Spanish

exponents of zero

exponents of zero

For all real numbers $\chi, \chi \neq 0, \chi^0=1$.
Negative Power Rule for Exponents

Negative Power Rule for Exponents

$\frac{1}{\chi^n} = \chi^{-n}$ where $\chi \neq 0$.
Negative Exponent Property

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

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

Zero Exponent Property

The zero exponent property says that for all $a \neq 0$, $a^0 = 1$.