Negative and Zero Exponent Rules

Learning Objectives

Define and use the zero exponent rule
Define and use the negative exponent rule

Define and use the zero exponent rule

When we defined the quotient rule, we only worked with expressions like the following: $\frac{{{4}^{9}}}{{{4}^{4}}}$ , where the exponent in the numerator (up) was greater than the one in the denominator (down), so the final exponent after simplifying was always a positive number, and greater than zero. In this section, we will explore what happens when we apply the quotient rule for exponents and get a negative or zero exponent.

What if the exponent is zero?

To see how this is defined, let us begin with an example. We will use the idea that dividing any number by itself gives a result of 1.

$\frac{t^{8}}{t^{8}}=\frac{\cancel{t^{8}}}{\cancel{t^{8}}}=1$

If we were to simplify the original expression using the quotient rule, we would have

$\frac{{t}^{8}}{{t}^{8}}={t}^{8 - 8}={t}^{0}$

If we equate the two answers, the result is ${t}^{0}=1$ . This is true for any nonzero real number, or any variable representing a real number.

a^{0} = 1

${a}^{0}=1$

The sole exception is the expression ${0}^{0}$ . This appears later in more advanced courses, but for now, we will consider the value to be undefined, or DNE (Does Not Exist).

Exponents of 0 or 1

Any number or variable raised to a power of 1 is the number itself.

$n^{1}=n$

Any non-zero number or variable raised to a power of 0 is equal to 1

$n^{0}=1$

The quantity $0^{0}$ is undefined.

As done previously, to evaluate expressions containing exponents of 0 or 1, substitute the value of the variable into the expression and simplify.

Example

Evaluate $2x^{0}$ if $x=9$

Show Solution

Substitute 9 for the variable x.

$2\cdot9^{0}$

Evaluate $9^{0}$ . Multiply.

$2\cdot1=2$

Answer

$2x^{0}=2$ , if $x=9$

Example

Simplify $\frac{{c}^{3}}{{c}^{3}}$ .

Show Solution

Use the quotient and zero exponent rules to simplify the expression.

$\begin{array}\text{ }\frac{c^{3}}{c^{3}} \,\,\,= \,\,\,c^{3-3} \\ \,\,\,\,\,\,\,\,\,\,\,=\,\,\,c^{0} \\\,\,\,\,\,\,\,\,\,\,\,=\,\,\,1\end{array}$

Answer

In the following video there is an example of evaluating an expression with an exponent of zero, as well as simplifying when you get a result of a zero exponent.

Define and use the negative exponent rule

We proposed another question at the beginning of this section. Given a quotient like $\displaystyle \frac{{{2}^{m}}}{{{2}^{n}}}$ what happens when n is larger than m? We will need to use the negative rule of exponents to simplify the expression so that it is easier to understand.

Let’s look at an example to clarify this idea. Given the expression:

$\frac{{h}^{3}}{{h}^{5}}$

Expand the numerator and denominator, all the terms in the numerator will cancel to 1, leaving two hs multiplied in the denominator, and a numerator of 1.

\begin{array}{l} \frac{h^{3}}{h^{5}} = \frac{h \cdot h \cdot h}{h \cdot h \cdot h \cdot h \cdot h} \\ = \frac{h \cdot h \cdot h}{h \cdot h \cdot h \cdot h \cdot h} \\ = \frac{1}{h \cdot h} \\ = \frac{1}{h^{2}} \end{array}

$\begin{array}{l} \frac{{h}^{3}}{{h}^{5}}\,\,\,=\,\,\,\frac{h\cdot{h}\cdot{h}}{h\cdot{h}\cdot{h}\cdot{h}\cdot{h}} \\ \,\,\,\,\,\,\,\,\,\,\,=\,\,\,\frac{\cancel{h}\cdot \cancel{h}\cdot \cancel{h}}{\cancel{h}\cdot \cancel{h}\cdot \cancel{h}\cdot {h}\cdot {h}}\\\,\,\,\,\,\,\,\,\,\,\,=\,\,\,\frac{1}{h\cdot{h}}\\\,\,\,\,\,\,\,\,\,\,\,=\,\,\,\frac{1}{{h}^{2}} \end{array}$

We could have also applied the quotient rule from the last section, to obtain the following result:

$\begin{array}{r}\frac{h^{3}}{h^{5}}\,\,\,=\,\,\,h^{3-5}\\\\=\,\,\,h^{-2}\,\,\end{array}$

Putting the answers together, we have ${h}^{-2}=\frac{1}{{h}^{2}}$ . This is true when h, or any variable, is a real number and is not zero.

The Negative Rule of Exponents

For any nonzero real number $a$ and natural number $n$ , the negative rule of exponents states that

a^{- n} = \frac{1}{a^{n}}

${a}^{-n}=\frac{1}{{a}^{n}}$

Let’s looks at some examples of how this rule applies under different circumstances.

Example

Evaluate the expression ${4}^{-3}$ .

Show Solution

First, write the expression with positive exponents by putting the term with the negative exponent in the denominator.

${4}^{-3} = \frac{1}{{4}^{3}} = \frac{1}{4\cdot4\cdot4}$

Now that we have an expression that looks somewhat familiar.

$\frac{1}{4\cdot4\cdot4} = \frac{1}{64}$

Answer

$\frac{1}{64}$

Example

Write $\frac{{\left({t}^{3}\right)}}{{\left({t}^{8}\right)}}$ with positive exponents.

Show Solution

Use the quotient rule to subtract the exponents of terms with like bases.

$\begin{array}{r}\frac{{\left({t}^{3}\right)}}{{\left({t}^{8}\right)}}={t}^{3-8}\\={t}^{-5}\,\,\end{array}$

Write the expression with positive exponents by putting the term with the negative exponent in the denominator.

$=\frac{1}{{t}^{5}}$

Answer

$\frac{1}{{t}^{5}}$

Example

Simplify ${\left(\frac{1}{3}\right)}^{-2}$ .

Show Solution

Apply the power property of exponents.

$\frac{{1}^{-2}}{{3}^{-2}}$

Write each term with a positive exponent, the numerator will go to the denominator and the denominator will go to the numerator.

$\frac{{3}^{2}}{{1}^{2}}{ = }\frac{{3}\cdot{3}}{{1}\cdot{1}}$

Simplify.

$\frac{{3}\cdot{3}}{{1}\cdot{1}}{ = }\frac{9}{1}{ = }{9}$

Answer

$9$

Example

Simplify. $\frac{1}{4^{-2}}$ Write your answer using positive exponents.

Show Solution

Write each term with a positive exponent, the denominator will go to the numerator.

$\frac{1}{4^{-2}}=1\cdot\frac{4^{2}}{1}=\frac{16}{1}=16$

Answer

In the follwoing video you will see examples of simplifying expressions with negative exponents.

Objective 3

Learning Objectives

Define and use the zero exponent rule

What if the exponent is zero?

Exponents of 0 or 1

Example

Answer

Example

Answer

Define and use the negative exponent rule

The Negative Rule of Exponents

Example

Answer

Example

Answer

Example

Answer

Example

Answer

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