## Negative and Zero Exponent Rules

### Learning Outcomes

• Simplify exponential expressions containing negative exponents and exponents of 0 and 1

## Define and Use the Zero Exponent Rule

Return to the quotient rule. We worked with expressions for which  $a>b$ so that the difference $a-b$ would never be zero or negative.

### The Quotient (Division) Rule for Exponents

For any non-zero number x and any integers a and b: $\displaystyle \frac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}$

What would happen if $a=b$? In this case, we would use the zero exponent rule of exponents to simplify the expression to $1$. To see how this is done, let us begin with an example.

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

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.

${a}^{0}=1$

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.

### The Zero Exponent Rule of Exponents

For any nonzero real number $a$, the zero exponent rule of exponents states that

${a}^{0}=1$

### Example

Simplify each expression using the zero exponent rule of exponents.

1. $\dfrac{{c}^{3}}{{c}^{3}}$
2. $\dfrac{-3{x}^{5}}{{x}^{5}}$
3. $\dfrac{{\left({j}^{2}k\right)}^{4}}{\left({j}^{2}k\right)\cdot {\left({j}^{2}k\right)}^{3}}$
4. $\dfrac{5{\left(r{s}^{2}\right)}^{2}}{{\left(r{s}^{2}\right)}^{2}}$

In the following video, you will see more examples of simplifying expressions whose exponents may be zero.

## Define and Use the Negative Exponent Rule

Another useful result occurs if we relax the condition that $a>b$ in the quotient rule even further. For example, can we simplify $\frac{{h}^{3}}{{h}^{5}}$? When $a<b$—that is, where the difference $a-b$ is negative—we can use the negative rule of exponents to simplify the expression to its reciprocal.

Divide one exponential expression by another with a larger exponent. Use our example, $\frac{{h}^{3}}{{h}^{5}}$.

$\Large\begin{array}{ccc}\hfill \frac{{h}^{3}}{{h}^{5}}& =& \frac{h\cdot h\cdot h}{h\cdot h\cdot h\cdot h\cdot h}\hfill \\ & =& \frac{\cancel{h}\cdot \cancel{h}\cdot \cancel{h}}{\cancel{h}\cdot \cancel{h}\cdot \cancel{h}\cdot h\cdot h}\hfill \\ & =& \frac{1}{h\cdot h}\hfill \\ & =& \frac{1}{{h}^{2}}\hfill \end{array}$

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

$\Large\begin{array}{ccc}\hfill \frac{{h}^{3}}{{h}^{5}}& =& {h}^{3 - 5}\hfill \\ & =& \text{ }{h}^{-2}\hfill \end{array}$

Putting the answers together, we have ${h}^{-2}=\frac{1}{{h}^{2}}$. This is true for any nonzero real number, or any variable representing a nonzero real number.

A factor with a negative exponent becomes the same factor with a positive exponent if it is moved across the fraction bar—from numerator to denominator or vice versa.

$\begin{array}{ccc}{a}^{-n}=\frac{1}{{a}^{n}}& \text{and}& {a}^{n}=\frac{1}{{a}^{-n}}\end{array}$

We have shown that the exponential expression ${a}^{n}$ is defined when $n$ is a natural number, $0$, or the negative of a natural number. That means that ${a}^{n}$ is defined for any integer $n$. Also, the product and quotient rules and all of the rules we will look at soon hold for any integer $n$.

### The Negative Rule of Exponents

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

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

### Example

Write each of the following quotients with a single base. Do not simplify further. Write answers with positive exponents.

1. $\dfrac{{(2b) }^{3}}{{(2b) }^{10}}$
2. $\dfrac{{z}^{2}\cdot z}{{z}^{4}}$
3. $\dfrac{{\left(-5{t}^{3}\right)}^{4}}{{\left(-5{t}^{3}\right)}^{8}}$

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

## Combine Exponent Rules to Simplify Expressions

Now we will combine the use of the product and quotient rules to simplify expressions whose terms may have negative or zero exponents.

### Example

Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents.

1. ${b}^{2}\cdot {b}^{-8}$
2. ${\left(-x\right)}^{5}\cdot {\left(-x\right)}^{-5}$
3. $\frac{-7z}{{\left(-7z\right)}^{5}}$

The following video shows more examples of how to combine the use of the product and quotient rules to simplify expressions whose terms may have negative or zero exponents.