Simplifying Expressions With Negative Exponents

Learning Outcomes

Use the properties of exponents to simplify products and quotients that contain negative exponents and variables

All the exponent properties we developed earlier in this chapter with whole number exponents apply to integer exponents, too. We restate them here for reference.

Summary of Exponent Properties

If [latex]a,b[/latex] are real numbers and [latex]m,n[/latex] are integers, then

[latex]\begin{array}{cccc}\mathbf{\text{Product Property}}\hfill & & & {a}^{m}\cdot {a}^{n}={a}^{m+n}\hfill \\ \mathbf{\text{Power Property}}\hfill & & & {\left({a}^{m}\right)}^{n}={a}^{m\cdot n}\hfill \\ \mathbf{\text{Product to a Power Property}}\hfill & & & {\left(ab\right)}^{m}={a}^{m}{b}^{m}\hfill \\ \mathbf{\text{Quotient Property}}\hfill & & & {\Large\frac{{a}^{m}}{{a}^{n}}}={a}^{m-n},a\ne 0\hfill \\ \mathbf{\text{Zero Exponent Property}}\hfill & & & {a}^{0}=1,a\ne 0\hfill \\ \mathbf{\text{Quotient to a Power Property}}\hfill & & & {\left({\Large\frac{a}{b}}\right)}^{m}={\Large\frac{{a}^{m}}{{b}^{m}}},b\ne 0\hfill \\ \mathbf{\text{Definition of Negative Exponent}}\hfill & & & {a}^{-n}={\Large\frac{1}{{a}^{n}}}\hfill \end{array}[/latex]

example

Simplify:

1. [latex]{x}^{-4}\cdot {x}^{6}[/latex]
2. [latex]{y}^{-6}\cdot {y}^{4}[/latex]
3. [latex]{z}^{-5}\cdot {z}^{-3}[/latex]

Solution

1.
	[latex]{x}^{-4}\cdot {x}^{6}[/latex]
Use the Product Property, [latex]{a}^{m}\cdot {a}^{n}={a}^{m+n}[/latex].	[latex]{x}^{-4+6}[/latex]
Simplify.	[latex]{x}^{2}[/latex]

2.
	[latex]{y}^{-6}\cdot {y}^{4}[/latex]
The bases are the same, so add the exponents.	[latex]{y}^{-6+4}[/latex]
Simplify.	[latex]{y}^{-2}[/latex]
Use the definition of a negative exponent, [latex]{a}^{-n}={\Large\frac{1}{{a}^{n}}}[/latex].	[latex]{\Large\frac{1}{{y}^{2}}}[/latex]

3.
	[latex]{z}^{-5}\cdot {z}^{-3}[/latex]
The bases are the same, so add the exponents.	[latex]{z}^{-5 - 3}[/latex]
Simplify.	[latex]{z}^{-8}[/latex]
Use the definition of a negative exponent, [latex]{a}^{-n}={\Large\frac{1}{{a}^{n}}}[/latex].	[latex]{\Large\frac{1}{{z}^{8}}}[/latex]

try it

In the next two examples, we’ll start by using the Commutative Property to group the same variables together. This makes it easier to identify the like bases before using the Product Property of Exponents.

example

Simplify: [latex]\left({m}^{4}{n}^{-3}\right)\left({m}^{-5}{n}^{-2}\right)[/latex]

Show Solution

try it

If the monomials have numerical coefficients, we multiply the coefficients, just as we did in Integer Exponents and Scientific Notation.

example

Simplify: [latex]\left(2{x}^{-6}{y}^{8}\right)\left(-5{x}^{5}{y}^{-3}\right)[/latex]

Show Solution

try it

In the next two examples, we’ll use the Power Property and the Product to a Power Property.

example

Simplify: [latex]{\left({k}^{3}\right)}^{-2}[/latex].

Show Solution

try it

example

Simplify: [latex]{\left(5{x}^{-3}\right)}^{2}[/latex]

Show Solution

try it

In the following video we show another example of how to simplify a product that contains negative exponents.

To simplify a fraction, we use the Quotient Property.

example

Simplify: [latex]{\Large\frac{{r}^{5}}{{r}^{-4}}}[/latex].

Show Solution

try it

In the next video we share more examples of simplifying a quotient with negative exponents.

	[latex]\left({m}^{4}{n}^{-3}\right)\left({m}^{-5}{n}^{-2}\right)[/latex]
Use the Commutative Property to get like bases together.	[latex]{m}^{4}{m}^{-5}\cdot {n}^{-2}{n}^{-3}[/latex]
Add the exponents for each base.	[latex]{m}^{-1}\cdot {n}^{-5}[/latex]
Take reciprocals and change the signs of the exponents.	[latex]{\Large\frac{1}{{m}^{1}}\cdot \frac{1}{{n}^{5}}}[/latex]
Simplify.	[latex]{\Large\frac{1}{m{n}^{5}}}[/latex]

	[latex]\left(2{x}^{-6}{y}^{8}\right)\left(-5{x}^{5}{y}^{-3}\right)[/latex]
Rewrite with the like bases together.	[latex]2\left(-5\right)\cdot \left({x}^{-6}{x}^{5}\right)\cdot \left({y}^{8}{y}^{-3}\right)[/latex]
Simplify.	[latex]-10\cdot {x}^{-1}\cdot {y}^{5}[/latex]
Use the definition of a negative exponent, [latex]{a}^{-n}={\Large\frac{1}{{a}^{n}}}[/latex].	[latex]-10\cdot {\Large\frac{1}{{x}^{1}}}\cdot {y}^{5}[/latex]
Simplify.	[latex]{\Large\frac{-10{y}^{5}}{x}}[/latex]

	[latex]{\left({k}^{3}\right)}^{-2}[/latex]
Use the Product to a Power Property, [latex]{\left(ab\right)}^{m}={a}^{m}{b}^{m}[/latex].	[latex]{k}^{3\left(-2\right)}[/latex]
Simplify.	[latex]{k}^{-6}[/latex]
Rewrite with a positive exponent.	[latex]{\Large\frac{1}{{k}^{6}}}[/latex]

	[latex]{\left(5{x}^{-3}\right)}^{2}[/latex]
Use the Product to a Power Property, [latex]{\left(ab\right)}^{m}={a}^{m}{b}^{m}[/latex].	[latex]{5}^{2}{\left({x}^{-3}\right)}^{2}[/latex]
Simplify [latex]{5}^{2}[/latex] and multiply the exponents of [latex]x[/latex] using the Power Property, [latex]{\left({a}^{m}\right)}^{n}={a}^{m\cdot n}[/latex].	[latex]25{x}^{-6}[/latex]
Rewrite [latex]{x}^{-6}[/latex] by using the definition of a negative exponent, [latex]{a}^{-n}={\Large\frac{1}{{a}^{n}}}[/latex].	[latex]25\cdot {\Large\frac{1}{{x}^{6}}}[/latex]
Simplify	[latex]{\Large\frac{25}{{x}^{6}}}[/latex]

	[latex]{\Large\frac{r^5}{r^{-4}}}[/latex]
Use the Quotient Property, [latex]{\Large\frac{{a}^{m}}{{a}^{n}}}={a}^{m-n}[/latex] .	[latex]{r}^{5-(\color{red}{-4})}[/latex]
Be careful to subtract [latex]5-(\color{red}{-4})[/latex]
Simplify.	[latex]r^9[/latex]

Module 1: Whole Numbers