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

### 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

#### Example A

*Simplify χ4χ4.*

**Solution:**

**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.*

**Solution:**

**Negative Power Rule for Exponents:**

#### Example C

*Rewrite using only positive exponents: χ−6γ−2.*

**Solution:**

#### Example D

*Write the following expressions without fractions.*

(a)

(b)

**Solution:**

(a)

(b)

Notice in part (a), the number 2 is in the numerator. This number is multiplied with

### Video Review

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### Guided Practice

Simplify

**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.

x−1⋅y2 x−4 x−3x−7 1x 2x2 x2y3 3xy 3x−3 a2b−3c−1 4x−1y3 2x−2y−3 (ab)−2 (3a−2b2c3)3 x−3⋅x3

Simplify the following expressions without any fractions in the answer.

a−3(a5)a−6 5x6y2x8y (4ab6)3(ab)5

Evaluate the following expressions to a single number.

3−2 (6.2)0 8−4⋅86

In 21 – 23, evaluate the expression for

2x2−3y3+4z (x2−y2)2 (3x2y54z)−2 - Evaluate \begin{align*}x^24x^3y^44y^2\end{align*} if \begin{align*}x=2\end{align*} and \begin{align*}y=-1\end{align*}.
- 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*}.
- 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*}.
- 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*}.
- Evaluate \begin{align*}3 \cdot 5^5 - 10 \cdot 5+1\end{align*}.
- Evaluate \begin{align*}\frac{2 \cdot 4^2-3 \cdot 5^2}{3^2}\end{align*}.
- 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.