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Negative Exponents
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What does it mean if our exponent is negative? Can you multiply a number to itself a negative amount of times? Instead of trying to wrap our brains around what that would mean, we use the exponent rule of division (quotient rule) to learn what a negative exponent means.

Rules of Exponents - Negative Exponents

As explained in the video, when we have a negative exponent we can simply move it to the other part of the fraction (from top to bottom or bottom to top) and then it will be a positive exponent. When doing your practice problems, remember you can use these rules in any order (product, quotient, and negative exponents) to simplify your expression. Many people like to use the negative exponent rule first because it’s less confusing to do the product and division rules once you don’t have any negative exponents.

### Practice Problems

1. Which expression is equivalent to $${\text{x}}^{-2}$$ ? (
Solution
Solution:
1.
Details:
The negative exponent rule says that a factor with a negative exponent can be moved to the other part of the fraction (top to bottom or bottom to top) and then have a positive exponent.

In this example, $${\text{x}}^{-2}$$ has a negative exponent of $$-2$$ , but it doesn’t look like a fraction. In cases like this, you must remember that anything divided by 1 is still itself. For example $$\dfrac{3}{1}=3$$ or $$12=\dfrac{12}{1}$$

We can apply this same principle here.

$${\text{x}}^{-2}=\dfrac{{\text{x}}^{-2}}{1}$$

Now it looks like a fraction.

The negative exponent rule now says we can do the following:

$${\text{x}}^{-2}=\dfrac{{\text{x}}^{-2}}{1} \rightarrow \dfrac{1}{{\text{x}}^{2}}$$

Our final answer is $$\dfrac{1}{{\text{x}}^{2}}$$
)
1. $$\dfrac{1}{{\text{x}}^{2}}$$
2. $$\dfrac{1}{{\text{x}}^{-2}}$$
3. $$\dfrac{1^{2}}{\text{x}}$$
4. $$\dfrac{2}{\text{x}}$$
2. Which expression is equivalent to $${\text{b}}^{-9}$$ ? (
Solution
Solution:
3.
)
1. $$\dfrac{1}{{\text{b}}^{-9}}$$
2. $$\dfrac{1^{9}}{\text{b}}$$
3. $$\dfrac{1}{{\text{b}}^{9}}$$
4. $$\dfrac{9}{\text{b}}$$
3. Write the expression $$\dfrac{{\text{y}}^{-7}}{{\text{y}}^{-13}}$$ using only positive exponents and simplify. (
Solution
Solution:
$${\text{y}}^{6}$$
Details:
Version 1: Use the negative exponent rule and then quotient rule

First, we apply the negative quotient rule that says as long as all the factors are being multiplied or divided together (no addition or subtraction) then we can move a factor with a negative exponent to the opposite side of a fraction and change the exponent to a positive.

Applying this we get:

$$\displaystyle\frac{{\text{y}}^{-7}}{{\text{y}}^{-13}}=\frac{{\text{y}}^{13}}{{\text{y}}^{7}}$$

Next, we apply the quotient rule that says as long as all the factors are being multiplied or divided (no addition or subtraction), and as long as factors have the same base, we can subtract the exponent of the factor in the denominator (bottom of the fraction) from the exponent of the factor in the numerator (top of the fraction).

$$\displaystyle\frac{{\text{y}}^{13}}{{\text{y}}^{7}}={\text{y}}^{\left ( 13-7 \right )}={\text{y}}^{6}$$

Version 2: Use the quotient rule and then the negative exponent rule

First, we apply the quotient rule. The quotient rule says that as long as the bases are the same, we can subtract the exponent of the factor in the denominator from the exponent of the factor in the numerator.

Applying this we get:

$$\dfrac{{\text{y}}^{-7}}{{\text{y}}^{-13}}={\text{y}}^{\left ( -7-\left ( -13 \right ) \right )}$$

$$\dfrac{{\text{y}}^{{\color{Blue} -7}}}{{\text{y}}^{{\color{DarkGreen} -13}}} = {\text{y}}^{({\color{Blue} -7}-\color{DarkGreen}(-13)\color{black})}$$

Subtracting a negative is the same as addition, so the exponent of y is:

$$(-7-(-13)) = -7 + 13$$

$$\dfrac{{\text{y}}^{{\color{Blue} -7}}}{{\text{y}}^{{\color{DarkGreen} -13}}} = {\text{y}}^{({\color{Blue} -7}-\color{DarkGreen}(-13)\color{black})} = {\text{y}}^{(-{\color{Blue} 7}+{\color{DarkGreen} 13})}$$

So $$-7 + 13 = 6$$ and our final answer is $${\text{y}}^{6}$$.

$$\dfrac{{\text{y}}^{{\color{Blue} -7}}}{{\text{y}}^{{\color{DarkGreen} -13}}} = {\text{y}}^{({\color{Blue} -7}-\color{DarkGreen}(-13)\color{black})} = {\text{y}}^{(-{\color{Blue} 7}+{\color{DarkGreen} 13})} = {\color{Red} {\text{y}}^{6}}$$
)
4. Write the expression $$\dfrac{{\text{m}}^{-5}{\text{x}}^{3}}{{\text{x}}^{-2}{\text{m}}^{-2}}$$ using only positive exponents and simplify. (
Solution
Solution:
$$\dfrac{{\text{x}}^{5}}{{\text{m}}^{3}}$$
)
5. Simplify the expression $$\dfrac{{\text{a}}^{7}{\text{b}}^{2}{\text{a}}^{-2}}{{\text{a}}^{-3}{\text{b}}^{7}}$$. (
Video Solution
Solution:
$$\dfrac{{\text{a}}^{8}}{{\text{b}}^{5}}$$
Details:

(Video Source | Transcript)
)
6. Simplify the expression $$\dfrac{{\text{x}}^{-5}{\text{y}}^{-3}{\text{z}}^{-2}}{{\text{x}}^{3}{\text{y}}^{-4}}$$. (
Video Solution
Solution:
$$\dfrac{\text{y}}{{\text{x}}^{8}{\text{z}}^{2}}$$
Details:

(Video Source | Transcript)
)

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