## Zero and Negative Exponents

### Learning Outcomes

• Simplify expressions with exponents equal to zero.
• Simplify expressions with negative exponents.
• Simplify exponential expressions.

Return to the quotient rule. We made the condition that $m>n$ so that the difference $m-n$ would never be zero or negative. What would happen if $m=n$? 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.

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

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

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

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.

### A General Note: The Zero Exponent Rule of Exponents

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

${a}^{0}=1$

### Example: Using the Zero Exponent Rule

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. $\dfrac{5{\left(r{s}^{2}\right)}^{2}}{{\left(r{s}^{2}\right)}^{2}}$

### Try It

Simplify each expression using the zero exponent rule of exponents.

1. $\dfrac{{t}^{7}}{{t}^{7}}$
2. $\dfrac{{\left(d{e}^{2}\right)}^{11}}{2{\left(d{e}^{2}\right)}^{11}}$
3. $\dfrac{{w}^{4}\cdot {w}^{2}}{{w}^{6}}$
4. $\dfrac{{t}^{3}\cdot {t}^{4}}{{t}^{2}\cdot {t}^{5}}$

In this video we show more examples of how to simplify expressions with zero exponents.

## Using the Negative Rule of Exponents

Another useful result occurs if we relax the condition that $m>n$ in the quotient rule even further. For example, can we simplify $\dfrac{{h}^{3}}{{h}^{5}}$? When $m<n$—that is, where the difference $m-n$ 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, $\dfrac{{h}^{3}}{{h}^{5}}$.

\begin{align} \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{align}

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

\begin{align} \frac{{h}^{3}}{{h}^{5}}& = {h}^{3 - 5} \\ & = {h}^{-2} \end{align}

Putting the answers together, we have ${h}^{-2}=\dfrac{1}{{h}^{2}}$. 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.

${a}^{-n}=\dfrac{1}{{a}^{n}} \text{ and } {a}^{n}=\dfrac{1}{{a}^{-n}}$

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 $n$.

### A General Note: The Negative Rule of Exponents

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

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

### Example: Using the Negative Exponent Rule

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

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

### Try It

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

1. $\dfrac{{\left(-3t\right)}^{2}}{{\left(-3t\right)}^{8}}$
2. $\dfrac{{f}^{47}}{{f}^{49}\cdot f}$
3. $\dfrac{2{k}^{4}}{5{k}^{7}}$

Watch this video to see more examples of simplifying expressions with negative exponents.

### Example: Using the Product and Quotient Rules

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. $\dfrac{-7z}{{\left(-7z\right)}^{5}}$

### Try It

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

1. ${t}^{-11}\cdot {t}^{6}$
2. $\dfrac{{25}^{12}}{{25}^{13}}$

## Finding the Power of a Product

To simplify the power of a product of two exponential expressions, we can use the power of a product rule of exponents, which breaks up the power of a product of factors into the product of the powers of the factors. For instance, consider ${\left(pq\right)}^{3}$. We begin by using the associative and commutative properties of multiplication to regroup the factors.

\begin{align} {\left(pq\right)}^{3}& = \stackrel{3\text{ factors}}{{\left(pq\right)\cdot \left(pq\right)\cdot \left(pq\right)}} \\ & = p\cdot q\cdot p\cdot q\cdot p\cdot q \\ & = \stackrel{3\text{ factors}}{{p\cdot p\cdot p}}\cdot \stackrel{3\text{ factors}}{{q\cdot q\cdot q}} \\ & = {p}^{3}\cdot {q}^{3} \end{align}

In other words, ${\left(pq\right)}^{3}={p}^{3}\cdot {q}^{3}$.

### A General Note: The Power of a Product Rule of Exponents

For any real numbers $a$ and $b$ and any integer $n$, the power of a product rule of exponents states that

$\large{\left(ab\right)}^{n}={a}^{n}{b}^{n}$

### Example: Using the Power of a Product Rule

Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.

1. ${\left(a{b}^{2}\right)}^{3}$
2. ${\left(2t\right)}^{15}$
3. ${\left(-2{w}^{3}\right)}^{3}$
4. $\dfrac{1}{{\left(-7z\right)}^{4}}$
5. ${\left({e}^{-2}{f}^{2}\right)}^{7}$

### Try It

Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.

1. ${\left({g}^{2}{h}^{3}\right)}^{5}$
2. ${\left(5t\right)}^{3}$
3. ${\left(-3{y}^{5}\right)}^{3}$
4. $\dfrac{1}{{\left({a}^{6}{b}^{7}\right)}^{3}}$
5. ${\left({r}^{3}{s}^{-2}\right)}^{4}$

In the following video we show more examples of how to find hte power of a product.

## Finding the Power of a Quotient

To simplify the power of a quotient of two expressions, we can use the power of a quotient rule, which states that the power of a quotient of factors is the quotient of the powers of the factors. For example, let’s look at the following example.

${\left({e}^{-2}{f}^{2}\right)}^{7}=\dfrac{{f}^{14}}{{e}^{14}}$

Let’s rewrite the original problem differently and look at the result.

\begin{align} {\left({e}^{-2}{f}^{2}\right)}^{7}& = {\left(\frac{{f}^{2}}{{e}^{2}}\right)}^{7} \\[1mm] & = \frac{{f}^{14}}{{e}^{14}} \\ \text{ } \end{align}

It appears from the last two steps that we can use the power of a product rule as a power of a quotient rule.

\begin{align} {\left({e}^{-2}{f}^{2}\right)}^{7}& = {\left(\frac{{f}^{2}}{{e}^{2}}\right)}^{7} \\[1mm] & = \frac{{\left({f}^{2}\right)}^{7}}{{\left({e}^{2}\right)}^{7}} \\[1mm] & = \frac{{f}^{2\cdot 7}}{{e}^{2\cdot 7}} \\[1mm] & = \frac{{f}^{14}}{{e}^{14}} \\ \text{ } \end{align}

### A General Note: The Power of a Quotient Rule of Exponents

For any real numbers $a$ and $b$ and any integer $n$, the power of a quotient rule of exponents states that

$\large{\left(\dfrac{a}{b}\right)}^{n}=\dfrac{{a}^{n}}{{b}^{n}}$

### Example: Using the Power of a Quotient Rule

Simplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with positive exponents.

1. ${\left(\dfrac{4}{{z}^{11}}\right)}^{3}$
2. ${\left(\dfrac{p}{{q}^{3}}\right)}^{6}$
3. ${\left(\dfrac{-1}{{t}^{2}}\right)}^{27}$
4. ${\left({j}^{3}{k}^{-2}\right)}^{4}$
5. ${\left({m}^{-2}{n}^{-2}\right)}^{3}$

### Try It

Simplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with positive exponents.

1. ${\left(\dfrac{{b}^{5}}{c}\right)}^{3}$
2. ${\left(\dfrac{5}{{u}^{8}}\right)}^{4}$
3. ${\left(\dfrac{-1}{{w}^{3}}\right)}^{35}$
4. ${\left({p}^{-4}{q}^{3}\right)}^{8}$
5. ${\left({c}^{-5}{d}^{-3}\right)}^{4}$

## Simplifying Exponential Expressions

Recall that to simplify an expression means to rewrite it by combing terms or exponents; in other words, to write the expression more simply with fewer terms. The rules for exponents may be combined to simplify expressions.

### Example: Simplifying Exponential Expressions

Simplify each expression and write the answer with positive exponents only.

1. ${\left(6{m}^{2}{n}^{-1}\right)}^{3}$
2. ${17}^{5}\cdot {17}^{-4}\cdot {17}^{-3}$
3. ${\left(\dfrac{{u}^{-1}v}{{v}^{-1}}\right)}^{2}$
4. $\left(-2{a}^{3}{b}^{-1}\right)\left(5{a}^{-2}{b}^{2}\right)$
5. ${\left({x}^{2}\sqrt{2}\right)}^{4}{\left({x}^{2}\sqrt{2}\right)}^{-4}$
6. $\dfrac{{\left(3{w}^{2}\right)}^{5}}{{\left(6{w}^{-2}\right)}^{2}}$

### Try It

Simplify each expression and write the answer with positive exponents only.

1. ${\left(2u{v}^{-2}\right)}^{-3}$
2. ${x}^{8}\cdot {x}^{-12}\cdot x$
3. ${\left(\frac{{e}^{2}{f}^{-3}}{{f}^{-1}}\right)}^{2}$
4. $\left(9{r}^{-5}{s}^{3}\right)\left(3{r}^{6}{s}^{-4}\right)$
5. ${\left(\frac{4}{9}t{w}^{-2}\right)}^{-3}{\left(\frac{4}{9}t{w}^{-2}\right)}^{3}$
6. $\dfrac{{\left(2{h}^{2}k\right)}^{4}}{{\left(7{h}^{-1}{k}^{2}\right)}^{2}}$

In the following video we show more examples of how to find the power of a quotient.

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