Negative Exponents

Introduction

In this lesson, you will learn about negative exponents.

What does it mean if an exponent is negative? Can you multiply a number to itself a negative amount of times? Instead of trying to figure that out, use the quotient rule of exponents to learn what a negative exponent means.

This video illustrates the lesson material below. Watching the video is optional

Rules of Exponents: Negative Exponents (07:39 mins) | Transcript

Negative Exponents

Example 1
Here is a base, b, raised to the negative exponent, negative 3, or written as \(b^{-3}\). What does this mean?

Review the rule for division to start. Think about \(\frac{a^{2}}{a^{5}}\). According to the quotient rule of exponents, \(\frac{a^{2}}{a^{5}}=a^{(2-5)}=a^{-3}\). What does this mean? What does a to the power of -3 mean? Doing this by hand looks like this:

\begin{align*}
\frac{a^{2}}{a^{5}}=\frac{\color{red} \cancel a \cdot \color{red} \cancel a}{ {\color{red} \cancel a \cdot \cancel a }\cdot a\cdot a \cdot a} = \frac{1}{a^{3}}
\end{align*}

In other words, \(\frac{1}{a^{3}}\) is the simplified answer where there is no negative exponent.

But remember, this can also be expressed as a to the negative 3, or \(a^{-3}\). So, any base raised to a negative exponent is actually equal to 1 over that number to the positive exponent, but in the denominator: \(\frac{1}{a^3} = a^{-3}\). Below is the rule for negative exponents.

\begin{align*}
\large\text{Negative Exponents Rule: } {x}^{-a} = \frac{1}{x^a} \\\\
\frac{{x}^{-a}}{1} = \frac{1}{x^a} \end{align*}

According to this rule, \(b^{-3} = \frac{1}{b^{3}}\).

Example 2
\(\frac{x^2}{x^4}\)

\begin{align*}
&\frac{x^2}{x^4} &\color{red}\small\text{Simplify this expression}\\\\
&x^{(2-4)} &\color{red}\small\text{Quotient rule of exponents}\\\\
&x^{-2} &\color{red}\small\text{Subtract exponents}\\\\
&\frac{1}{x^2} &\color{red}\small\text{Negative exponent rule}
\end{align*}

\(x^{-2}\) can also be expressed as \(\frac{1}{x^{2}}\).

Example 3
Simplify this expression: \(\frac{b^{-1}a^{-2}}{ab}\). Use the Quotient Rule of Exponents and the Negative Exponent Rule.

Applying the Quotient Rule of Exponents first:

\begin{align*}&\frac{b^{-1}a^{-2}}{ab}&\color\red\small\text{Simplify this expression}\\\\ &\frac{a^{-2}b^{-1}}{ab}&\color\red\small\text{Rearrange the order}\\\\ &a^{(-2-1)}b^{(-1-1)} &\color\red\small\text{Quotient rule of exponents}\\\\&a^{-3}b^{-2} &\color\red\small\text{Subtract the exponents}\\\\&\frac{1}{a^{3}b^{2}} &\color\red\small\text{Negative exponent rule}\\\\\end{align*}

Applying the Negative Exponents Rule first:

\begin{align*}&\frac{b^{-1}a^{-2}}{ab}&\color\red\small\text{Simplify this expression}\\\\&\frac{1}{a^{1}a^{2}b^{1}b^{1}} &\color\red\small\text{Negative exponent rule}\\\\&\frac{1}{a^{(1+2)}b^{(1+1)}} &\color\red\small\text{Product rule of exponents}\\\\ &\frac{1}{a^{3}b^{2}} &\color\red\small\text{Add the exponents}
\end{align*}

Either method gets the same results, \(\frac{1}{a^{3}b^{2}}\).

A simplified way to think about a term with a negative exponent is that it’s just on the opposite side of the division line. For example:

\begin{align*}a^{-3}& =\frac{a^{-3}}{1} = \frac{1}{a^3}\\\\&\text{and similarly,}\\\\&\frac{1}{a^{-3}} = a^{3}\\\\\end{align*}

Example 4
Simplify this expression: \(\frac{b^{-2}a^{-3}}{c^{-4}d^{-5}}\).

According to the rule of negative exponents (if you move everything that’s negative onto the other side of the division line) it’s equal to \(\frac{c^{4}d^{5}}{b^{2}a^{3}}\). Everything just moved from one side of the division line to the other.

When you have a negative exponent, you 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.

You can use these rules in any order (product, quotient, and negative exponents) to simplify an 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.

Things to Remember

The rule of negative exponents states that any number or variable to a negative exponent is the same as one over that same number or variable to the positive exponent. \(b^{-5} = \frac{1}{b^{5}}\)
When you move a negative exponent (with its base) from one side of the division line to the other (numerator to denominator or denominator to numerator), the exponents will then be positive. (See Example 4.)
Make sure to check your work! It is important that you go back through again on each problem to check the exponents are correct.

Practice Problems

Which expression is equivalent to \({\text{x}}^{-2}\) ? (
Solution

x

Solution: a. \(\dfrac{1}{{\text{x}}^{2}}\)
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 \(12=\dfrac{12}{1}\).

You 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 you can do the following:

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

The 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. \(\dfrac{2}{\text{x}}\)
Which expression is equivalent to \({\text{b}}^{-9}\) ? (
Solution

x

Solution: c. \(\dfrac{1}{{\text{b}}^{9}}\)

)
1. \(\dfrac{1}{{\text{b}}^{-9}}\)
2. \(\dfrac{1^{9}}{\text{b}}\)
3. \(\dfrac{1}{{\text{b}}^{9}}\)
4. \(\dfrac{9}{\text{b}}\)
Write the expression \(\dfrac{{\text{y}}^{-7}}{{\text{y}}^{-13}}\) using only positive exponents and simplify. (
Solution

x

Solution: \({\text{y}}^{6}\)
Details:
Version 1: Use the negative exponent rule and then quotient rule

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

Applying this you get:

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

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

Applying this you 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:

\((-7-(-13)) = -7 + 13\)

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

Therefore, \(-7 + 13 = 6\) and the final answer is: \({\text{y}}^{6}\).

\(\dfrac{{\text{y}}^{{\color{Blue} -7}}}{{\text{y}}^{{\color{DarkGreen} -13}}} = {\text{y}}^{({\color{Blue} -7}-\color{DarkGreen}(-13)\color{black})} = {\text{y}}^{(-{\color{Blue} 7}+{\color{DarkGreen} 13})} = {\color{Red} {\text{y}}^{6}}\)

)
Write the expression \(\dfrac{{\text{m}}^{-5}{\text{x}}^{3}}{{\text{x}}^{-2}{\text{m}}^{-2}}\) using only positive exponents and simplify. (
Solution

x

Solution: \(\dfrac{{\text{x}}^{5}}{{\text{m}}^{3}}\)

)
Simplify the expression \(\dfrac{{\text{a}}^{7}{\text{b}}^{2}{\text{a}}^{-2}}{{\text{a}}^{-3}{\text{b}}^{7}}\). (
Video Solution

x

Solution: \(\dfrac{{\text{a}}^{8}}{{\text{b}}^{5}}\)
Details:

(Negative Exponents #5 (05:13 mins) | Transcript)

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

x

Solution: \(\dfrac{\text{y}}{{\text{x}}^{8}{\text{z}}^{2}}\)
Details:

(Negative Exponents #6 (04:45 mins) | Transcript)

| Transcript)

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