2.6 Integer Exponents

Leanne Thompson

2.6 Integer Exponents

In this section, we will begin to work with negative integer exponents. As well as continuing to use the exponent laws from previous lessons, we will also begin to use the Negative Exponent Law. Work through the following exercises to derive the Negative Exponent Law.

Simplify $\frac{{x}^{2}}{{x}^{5}}$ using the Quotient Law.

Now simplify the same expression, $\frac{{x}^{2}}{{x}^{5}}$ , by writing the exponents in expanded form and then dividing out common factors.

This implies that = , which leads us to the definition of the Negative Exponent Law.

Negative Exponent Law

If $n$ is an integer and $a\ne 0$ , then ${a}^{-n}=\frac{1}{{a}^{n}}$ .

The Negative Exponent Law tells us that we can rewrite an expression by taking the reciprocal of the base, then changing the sign of the exponent. Any expression that has negative exponents is not considered to be in simplest form. We will use the definition of a negative exponent and other properties of exponents to write the expression with only positive exponents. For example, if after simplifying an expression, we end up with the expression ${x}^{-3}$ , we will take one more step and write $\frac{1}{{x}^{3}}$ . The answer is considered to be in simplest form when it has only positive exponents.

Example 1

Simplify ${4}^{-2}$ .

Table 2.6.1
Steps	Solution
Use the Negative Exponent Law: ${a}^{-n}=\frac{1}{{a}^{n}}$ .	${4}^{-2}$ $\frac{1}{{4}^{2}}$
Simplify.	$\frac{1}{16}$

Practice 1

Simplify.

a) ${2}^{-3}$

b) ${3}^{-2}$

c) ${10}^{-4}$

In Example 1, we raised an integer to a negative exponent. We will now look at what happens when we raise a fraction to a negative exponent. To explore this, first look at the work below to see what happens to a fraction whose numerator is one and denominator is an integer raised to a negative exponent.

Table 2.6.2
Steps	Reason
$\frac{1}{{a}^{-n}}$ $\frac{1}{\frac{1}{{a}^{n}}}$	Use the Negative Exponent Law: ${a}^{-n}=\frac{1}{{a}^{n}}$ .
$1 \bullet \frac{{a}^{n}}{1}$	Simplify the complex fraction.
${a}^{n}$	Multiply.

This leads us to another exponent law that we can use when working with negative exponents:

If $n$ is an integer and $a\ne 0$ , then $\frac{1}{{a}^{-n}}={a}^{n}$ .

Example 2

Simplify $\frac{1}{{y}^{-4}}$ .

Table 2.6.3
Steps	Solution
Use the Negative Exponent Law: $\frac{1}{{a}^{-n}}={a}^{n}$ .	$\frac{1}{{y}^{-4}}$ ${y}^{4}$

Practice 2

Simplify. Make sure you express each answer with positive exponents.

a) $\frac{1}{{m}^{-8}}$

b) $\frac{p}{{q}^{-7}}$

c) $\frac{1}{{4}^{-3}}$

d) $\frac{1}{{2}^{-4}}$

Suppose we now have a fraction raised to a negative exponent. Let’s use the Negative Exponent Law to lead us to a new law.

Table 2.6.4
Steps	Reason
${\left(\frac{3}{4}\right)}^{-2}$ $\frac{1}{{\left(\frac{3}{4}\right)}^{2}}$	Use the Negative Exponent Law: ${a}^{-n}=\frac{1}{{a}^{n}}$ .
$\frac{1}{\frac{9}{16}}$	Simplify the denominator.
$\frac{16}{9}$	Simplify the complex fraction.
$\frac{16}{9}$ = ${\left(\frac{4}{3}\right)}^{2}$	But we know that $\frac{16}{9}$ is ${\left(\frac{4}{3}\right)}^{2}$ .
${\left(\frac{3}{4}\right)}^{-2}={\left(\frac{4}{3}\right)}^{2}$	To make the exponent positive, we can take the reciprocal of the base and change the sign of the exponent.

To get from the original fraction raised to a negative exponent to the final result, we took the reciprocal of the base—the fraction—and changed the sign of the exponent. This leads us to another exponent law that we can use when working with negative exponents:

If $a\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b$ are real numbers, $a\ne 0,b\ne 0$ , and $n$ is an integer, then ${\left(\frac{a}{b}\right)}^{-n}={\left(\frac{b}{a}\right)}^{n}$ .

Example 3

Simplify ${\left(\frac{5}{7}\right)}^{-2}$ .

Table 2.6.5
Steps	Solution
Use the Negative Exponent Law: ${\left(\frac{a}{b}\right)}^{-n}={\left(\frac{b}{a}\right)}^{n}$ . To do this take the reciprocal of the fraction and change the sign of the exponent.	${\left(\frac{5}{7}\right)}^{-2}$ ${\left(\frac{7}{5}\right)}^{2}$
Simplify.	$\frac{49}{25}$

Practice 3

Simplify.

a) ${\left(\frac{2}{3}\right)}^{-4}$

b) ${\left(\frac{3}{5}\right)}^{-3}$

Practice 4

Use exponent laws to simplify. Make sure you express each answer with positive exponents.

a) $5{x}^{-2}$

b) ${\left(5x\right)}^{-2}$

Practice 5

Use exponent laws to simplify. Make sure you express each answer with positive exponents.

a) ${y}^{-7} \bullet {y}^{2}$

b) ${\left(6{k}^{3}\right)}^{-2}$

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

d) $\frac{{(3m)}^{-2}}{{m}^{5}}$

Practice 6

Use exponent laws to simplify. Make sure you express each answer with positive exponents.

a) $\frac{18{x}^{-3}{y}^{2}}{6{x}^{2}{y}^{-4}}$

b) $\left(\frac{-6{p}^{5}{q}^{3}}{12{p}{q}^{-6}}\right)^{-2}$

Homework

Evaluate. Write your answers as exact values.

a) ${3}^{-4}$

b) ${10}^{-2}$

c) ${2}^{-8}$

d) ${10}^{-5}$
Simplify. Write your answers with positive exponents.

a) $\frac{1}{{c}^{-5}}$

b) $\frac{1}{{5}^{-2}}$

c) $\frac{{u}^{5}}{{t}^{-9}}$

d) $\frac{1}{{10}^{-4}}$
Simplify. Write your answers with positive exponents.

a) ${\left(\frac{3}{10}\right)}^{-2}$

b) ${\left(\frac{7}{2}\right)}^{-3}$
Evaluate. Write your answers as exact values.

a) ${\left(-7\right)}^{-2}$

b) $-{7}^{-2}$

c) ${\left(-\frac{1}{7}\right)}^{-2}$

d) $-{\left(\frac{1}{7}\right)}^{-2}$

e) $-{5}^{-3}$

f) ${\left(-\frac{1}{5}\right)}^{-3}$

g) $-{\left(\frac{1}{5}\right)}^{-3}$

h) ${\left(-5\right)}^{-3}$
Simplify. Write your answers with positive exponents.

a) ${s}^{3} \bullet {s}^{-7}$

b) ${q}^{-8} \bullet {q}^{3}$

c) ${y}^{-2} \bullet {y}^{-5}$

d) ${y}^{5} \bullet {y}^{-5}$

e) $y \bullet {y}^{5}$

f) $y \bullet {y}^{-5}$

g) ${x}^{4} \bullet {x}^{-2} \bullet {x}^{-3}$

h) $\left({m}^{3}{n}^{-3}\right)\left({m}^{-5}{n}^{-1}\right)$

i) $\left(p{q}^{-4}\right)\left({p}^{-6}{q}^{-3}\right)$

j) $\left(-2{j}^{-5}{k}^{8}\right)\left(7{j}^{2}{k}^{-3}\right)$

k) $\left(-5{m}^{4}{n}^{6}\right)\left(8{m}^{-5}{n}^{-3}\right)$

l) ${\left(4{y}^{3}\right)}^{-3}$

m) ${\left(2{p}^{-5}\right)}^{2}$

n) $\frac{{n}^{5}}{{n}^{-2}}$

o) $\frac{{y}^{-5}}{{y}^{-10}}$
Simplify. Write your answers with positive exponents.

a) $\frac{{p}^{-2}{q}^{-5}}{{p}^{3}{q}^{-4}}$

b) $\left(\frac{3{x}^{2}{y}^{4}}{15{x}{y}^{-2}}\right)^{-2}$

c) $\left(\dfrac{x^{-2}y^{-6}}{x^{-2}y^4}\right)^2$

d) $\left(\dfrac{x^{-3}y^{-3}}{x^{-1}y^6}\right)^3$

e) $\left(\dfrac{x^{-2}y^{-4}}{x^2y^{-4}}\right)^2$

f) $\left(\dfrac{x^{-5}y^{-3}}{x^{-4}y^2}\right)^4$

Answers

1.

a) $\frac{1}{81}$

b) $\frac{1}{100}$

c) $\frac{1}{256}$

d) $\frac{1}{100 000}$

2.

a) ${c}^{5}$

b) ${5}^{2}$ = 25

c) ${t}^{9}{u}^{5}$

d) ${10}^{4}$ = 10 000

3.

a) $\frac{100}{9}$

b) $\frac{8}{343}$

4.

a) $\frac{1}{49}$	b) $-\frac{1}{49}$	c) 49	d) $-49$
e) $-\frac{1}{125}$	f) $-125$	g) $-125$	h) $-\frac{1}{125}$

5.

a) $\frac{1}{{s}^{4}}$	b) $\frac{1}{{q}^{5}}$	c) $\frac{1}{{y}^{7}}$	d) 1	e) ${y}^{6}$
f) $\frac{1}{{y}^{4}}$	g) $\frac{1}{x}$	h) $\frac{1}{{m}^{2}{n}^{4}}$	i) $\frac{1}{{p}^{5}{q}^{7}}$	j) $-\frac{14{k}^{5}}{{j}^{3}}$
k) $-\frac{40{n}^{3}}{m}$	l) $\frac{1}{64{y}^{9}}$	m) $\frac{4}{{p}^{10}}$	n) ${n}^{7}$	o) ${y}^{5}$