Simplifying Complex Expressions I

Learning Outcomes

Simplify complex expressions using a combination of exponent rules
Simplify quotients that require a combination of the properties of exponents

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 as we will be using them here to simplify various exponential expressions.

Summary of Exponent Properties

If $a,b$ are real numbers and $m,n$ are integers, then

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

Expressions with negative exponents

The following examples involve simplifying expressions with negative exponents.

example

Simplify:

1. ${x}^{-4}\cdot {x}^{6}$
2. ${y}^{-6}\cdot {y}^{4}$
3. ${z}^{-5}\cdot {z}^{-3}$

Solution

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

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

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

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: $\left({m}^{4}{n}^{-3}\right)\left({m}^{-5}{n}^{-2}\right)$

Show Solution

Solution

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

try it

If we multipy two expressions with numerical coefficients, we multiply the coefficients together.

example

Simplify: $\left(2{x}^{-6}{y}^{8}\right)\left(-5{x}^{5}{y}^{-3}\right)$

Show Solution

Solution

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

try it

In the next two examples, we’ll use the Power Property and the Product to a Power Property to simplify expressions with negative exponents.

example

Simplify: ${\left({k}^{3}\right)}^{-2}$ .

Show Solution

Solution

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

try it

example

Simplify: ${\left(5{x}^{-3}\right)}^{2}$

Show Solution

Solution

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

try it

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

The following examples involve solving exponential expressions with quotients.

example

Simplify: ${\Large\frac{{\left({x}^{2}\right)}^{3}}{{x}^{5}}}$ .

Solution

\frac{{(x^{2})}^{3}}{x^{5}}

${\Large\frac{{\left({x}^{2}\right)}^{3}}{{x}^{5}}}$

Multiply the exponents in the numerator, using the

Power Property.

\frac{x^{6}}{x^{5}}

${\Large\frac{{x}^{6}}{{x}^{5}}}$

Subtract the exponents.

x

$x$

try it

example

Simplify: ${\Large\frac{{m}^{8}}{{\left({m}^{2}\right)}^{4}}}$

Show Solution

Solution

\frac{m^{8}}{{(m^{2})}^{4}}

${\Large\frac{{m}^{8}}{{\left({m}^{2}\right)}^{4}}}$

Multiply the exponents in the numerator, using the

Power Property.

\frac{m^{8}}{m^{8}}

${\Large\frac{{m}^{8}}{{m}^{8}}}$

Subtract the exponents.

m^{0} = 1

${m}^{0}=1$

try it

example

Simplify: ${\left({\Large\frac{{x}^{7}}{{x}^{3}}}\right)}^{2}$

Show Solution

Solution

	${\left(\frac{{x}^{7}}{{x}^{3}}\right)}^{2}$
Remember parentheses come before exponents, and the bases are the same so we can simplify inside the parentheses. Subtract the exponents.	${\left({x}^{7 - 3}\right)}^{2}$
Simplify.	${\left({x}^{4}\right)}^{2}$
Multiply the exponents.	${x}^{8}$

try it

example

Simplify: ${\left({\Large\frac{{p}^{2}}{{q}^{5}}}\right)}^{3}$

Show Solution

Solution
Here we cannot simplify inside the parentheses first, since the bases are not the same.

	${\Large{\left(\frac{{p}^{2}}{{q}^{5}}\right)}}^{3}$
Raise the numerator and denominator to the third power using the Quotient to a Power Property, ${\Large{\left(\frac{a}{b}\right)}}^{m}={\Large\frac{{a}^{m}}{{b}^{m}}}$	${\Large\frac{(p^2)^{3}}{(q^5)^{3}}}$
Use the Power Property, ${\left({a}^{m}\right)}^{n}={a}^{m\cdot n}$ .	${\Large\frac{{p}^{6}}{{q}^{15}}}$

try it

example

Simplify: ${\Large{\left(\frac{2{x}^{3}}{3y}\right)}}^{4}$

Show Solution

Solution

	${\Large{\left(\frac{2{x}^{3}}{3y}\right)}}^{4}$
Raise the numerator and denominator to the fourth power using the Quotient to a Power Property.	${\Large\frac{{\left(2{x}^{3}\right)}^{4}}{{\left(3y\right)}^{4}}}$
Raise each factor to the fourth power, using the Power to a Power Property.	${\Large\frac{{2}^{4}{\left({x}^{3}\right)}^{4}}{{3}^{4}{y}^{4}}}$
Use the Power Property and simplify.	${\Large\frac{16{x}^{12}}{81{y}^{4}}}$

try it

example

Simplify: ${\Large\frac{{\left({y}^{2}\right)}^{3}{\left({y}^{2}\right)}^{4}}{{\left({y}^{5}\right)}^{4}}}$

Show Solution

Solution

	${\Large\frac{{\left({y}^{2}\right)}^{3}{\left({y}^{2}\right)}^{4}}{{\left({y}^{5}\right)}^{4}}}$
Use the Power Property.	${\Large\frac{\left({y}^{6}\right)\left({y}^{8}\right)}{{y}^{20}}}$
Add the exponents in the numerator, using the Product Property.	${\Large\frac{{y}^{14}}{{y}^{20}}}$
Use the Quotient Property.	${\Large\frac{1}{{y}^{6}}}$

try it

For more similar examples, watch the following video.

To conclude this section, we will simplify quotient expressions with a negative exponent.

example

Simplify: ${\Large\frac{{r}^{5}}{{r}^{-4}}}$ .

Show Solution

Solution

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

try it

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

Module 12: Exponents

Learning Outcomes

Summary of Exponent Properties

Expressions with negative exponents

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