Zero and Negative Exponents (With Examples)

Zero and negative exponents follow two rules. Any nonzero number to the zero power is 1, so $7^0 = 1$ . A negative exponent means “one over” the positive power, so $x^{-n} = \dfrac{1}{x^n}$ . For example, $2^{-3} = \dfrac{1}{8}$ . A negative exponent never makes the number negative.

Before this lesson: You should know the exponent rules, what exponents mean, and how to work with fractions.

Extra practice: The Algebra 1 Unit 6 Practice Pack has printable problems, answer keys and a cheat sheet for this unit.

Why Zero and Negative Exponents Work

Look at the powers of 2. Each time the exponent goes down by 1, you divide by 2. So $2^3 = 8$ , $2^2 = 4$ and $2^1 = 2$ .

Keep going. Divide 2 by 2 and you get 1, so $2^0 = 1$ . Divide again and you get $\tfrac{1}{2}$ , so $2^{-1} = \tfrac{1}{2}$ . Then $2^{-2} = \tfrac{1}{4}$ .

Each step to the right lowers the exponent by 1 and divides by 2. The pattern gives 2⁰ = 1 and 2⁻¹ = 1/2.

The quotient rule says the same thing. $\dfrac{x^3}{x^3} = x^{3-3} = x^0$ . But anything divided by itself is 1. So $x^0$ has to be 1.

Zero and Negative Exponent Rules

In each rule, $x$ is not 0, and $n$ is a whole number.

Rule	In symbols	Example
Zero exponent	$x^0 = 1$	$9^0 = 1$
Negative exponent	$x^{-n} = \dfrac{1}{x^n}$	$5^{-2} = \dfrac{1}{25}$
Negative exponent on the bottom	$\dfrac{1}{x^{-n}} = x^n$	$\dfrac{1}{3^{-2}} = 9$
Fraction to a negative power	$\left(\dfrac{a}{b}\right)^{-n} = \left(\dfrac{b}{a}\right)^n$	$\left(\dfrac{2}{3}\right)^{-2} = \dfrac{9}{4}$

A negative exponent moves a factor across the fraction bar. A factor on top moves to the bottom. A factor on the bottom moves to the top. The exponent turns positive when it moves.

How to Rewrite with Positive Exponents

Find each factor with a zero exponent. Change it to 1.
Find each factor with a negative exponent. Look at what the exponent is attached to.
Move only that factor across the fraction bar, and make its exponent positive.
Leave every other factor where it is, including numbers with no exponent.
Use the exponent rules to finish. Simplify any numbers.

Worked Examples

Example 1: Evaluate Powers

Find the value of (a) $6^0$ , (b) $(-4)^0$ , (c) $-4^0$ and (d) $3^{-2}$ .

6^0 = 1

Step 1. (a) Any nonzero number to the zero power is 1.

(-4)^0 = 1

Step 2. (b) The base is $-4$ , inside the parentheses. It is a nonzero number, so the power is 1.

-4^0 = -(4^0)

Step 3. (c) There are no parentheses, so the base is just 4. The minus sign stays out front.

-(4^0) = -1

Step 4. Find $4^0 = 1$ .

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

Step 5. (d) Move the power to the bottom of a fraction and make the exponent positive.

\frac{1}{3^2} = \frac{1}{9}

Step 6. Find $3^2 = 9$ . (See exponents.)

Answer: (a) 1; (b) 1; (c) $-1$ ; (d) $\dfrac{1}{9}$

Example 2: Watch What the Exponent Is Attached To

Rewrite with positive exponents: (a) $x^{-5}$ , (b) $4y^{-2}$ and (c) $(4y)^{-2}$ .

x^{-5} = \frac{1}{x^5}

Step 1. (a) Move the power to the bottom and make the exponent positive.

4y^{-2} = 4 \cdot \frac{1}{y^2}

Step 2. (b) The exponent is on $y$ only. Rewrite $y^{-2}$ and keep the 4 on top.

4 \cdot \frac{1}{y^2} = \frac{4}{y^2}

Step 3. Multiply.

(4y)^{-2} = \frac{1}{(4y)^2}

Step 4. (c) Now the exponent is on the whole group $4y$ . Move the group to the bottom.

\frac{1}{(4y)^2} = \frac{1}{16y^2}

Step 5. Square both factors: $(4y)^2 = 4^2y^2 = 16y^2$ .

Answer: (a) $\dfrac{1}{x^5}$ ; (b) $\dfrac{4}{y^2}$ ; (c) $\dfrac{1}{16y^2}$

Example 3: Negative Exponents on the Bottom

Rewrite with positive exponents: (a) $\dfrac{5}{a^{-3}}$ and (b) $\dfrac{m^{-2}}{n^{-4}}$ .

\frac{5}{a^{-3}} = 5a^3

Step 1. (a) The factor $a^{-3}$ is on the bottom. Move it to the top and make the exponent positive.

\frac{m^{-2}}{n^{-4}} = \frac{1}{m^2 n^{-4}}

Step 2. (b) Move $m^{-2}$ down. Its exponent turns positive.

\frac{1}{m^2 n^{-4}} = \frac{n^4}{m^2}

Step 3. Move $n^{-4}$ up. Its exponent turns positive too.

Answer: (a) $5a^3$ ; (b) $\dfrac{n^4}{m^2}$

Example 4: Simplify with the Exponent Rules

Simplify. Write the answer with positive exponents only: (a) $(2x^{-3}y^4)(5x^7y^{-6})$ and (b) $\dfrac{12a^2b^{-1}}{3a^5b^2}$ .

2 \cdot 5 = 10

Step 1. (a) Multiply the numbers.

x^{-3} \cdot x^7 = x^{-3+7} = x^4

Step 2. Add the exponents on $x$ (see exponent rules).

y^4 \cdot y^{-6} = y^{4+(-6)} = y^{-2}

Step 3. Add the exponents on $y$ .

10x^4y^{-2}

Step 4. Put the parts together.

10x^4y^{-2} = \frac{10x^4}{y^2}

Step 5. Move $y^{-2}$ to the bottom and make the exponent positive.

\frac{12}{3} = 4

Step 6. (b) Divide the numbers.

\frac{a^2}{a^5} = a^{2-5} = a^{-3}

Step 7. Subtract the exponents on $a$ .

\frac{b^{-1}}{b^2} = b^{-1-2} = b^{-3}

Step 8. Subtract the exponents on $b$ .

4a^{-3}b^{-3}

Step 9. Put the parts together.

4a^{-3}b^{-3} = \frac{4}{a^3b^3}

Step 10. Move both negative powers to the bottom.

Answer: (a) $\dfrac{10x^4}{y^2}$ ; (b) $\dfrac{4}{a^3b^3}$

Example 5: Word Problem

A ladybug is 4 millimeters long. One millimeter is $10^{-3}$ meters. (a) Write the ladybug’s length in meters. (b) How many ladybugs, lined up end to end, would make 2 meters?

4 \cdot 10^{-3}

Step 1. (a) Multiply the length in millimeters by the size of one millimeter.

4 \cdot \frac{1}{1000} = \frac{4}{1000}

Step 2. Rewrite the negative exponent as a fraction.

\frac{4}{1000} = 0.004

Step 3. Write the fraction as a decimal (see converting fractions and decimals).

\frac{2}{4 \cdot 10^{-3}}

Step 4. (b) Divide the total length by the length of one ladybug.

\frac{2}{4} \cdot \frac{1}{10^{-3}}

Step 5. Split the fraction into two parts.

\frac{1}{2} \cdot 10^3

Step 6. Simplify $\frac{2}{4}$ and move $10^{-3}$ to the top.

\frac{1}{2} \cdot 1000 = 500

Step 7. Find $10^3 = 1000$ and multiply.

Answer: (a) 0.004 meters; (b) 500 ladybugs

Check: $500 \cdot 0.004 = 2$ meters.

Common Mistakes

Mistake	Example of the error	Fix
Making the number negative	$2^{-3} = -8$	A negative exponent means a fraction: $2^{-3} = \dfrac{1}{8}$
Multiplying by the exponent	$2^{-3} = -6$	The exponent is not a factor: $2^{-3} = \dfrac{1}{8}$
Moving the coefficient too	$3x^{-2} = \dfrac{1}{3x^2}$	Only $x$ has the exponent: $\dfrac{3}{x^2}$
Saying the zero power is 0	$8^0 = 0$	$8^0 = 1$
Ignoring a minus sign out front	$-5^0 = 1$	The base is 5: $-5^0 = -1$ . But $(-5)^0 = 1$
Leaving a negative exponent in the answer	$\dfrac{a^{-2}}{b}$	Move it down: $\dfrac{1}{a^2b}$

Negative Exponents and Powers of 10

Negative powers of 10 are small decimals. The exponent tells you how many places the 1 sits after the decimal point.

Power	Fraction	Decimal
$10^2$	$100$	100
$10^1$	$10$	10
$10^0$	$1$	1
$10^{-1}$	$\dfrac{1}{10}$	0.1
$10^{-2}$	$\dfrac{1}{100}$	0.01
$10^{-3}$	$\dfrac{1}{1000}$	0.001

This is how scientific notation writes very small numbers. For example, $7.2 \cdot 10^{-4} = 0.00072$ . Metric units work the same way: a milligram is $10^{-3}$ grams.

Practice Problems

Find the value of $15^0$ .
Find the value of $4^{-3}$ .
Find the value of $-2^{-2}$ .
Rewrite $7m^{-4}$ with positive exponents.
Rewrite $\dfrac{3}{k^{-2}}$ with positive exponents.
Find the value of $\left(\dfrac{3}{5}\right)^{-2}$ .
Simplify $x^{-6} \cdot x^{9}$ .
Simplify $\dfrac{20p^3q^{-2}}{5p^7}$ . Use positive exponents only.
Simplify $(3a^{-2}b)^2 \cdot a^5$ . Use positive exponents only.
A dust grain has a mass of $5 \cdot 10^{-6}$ grams. Write this as a decimal. Then find the mass of 1,000 grains.

Show answers

15^0 = 1

Any nonzero number to the zero power is 1.

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

Move the power to the bottom and make the exponent positive.

4^3 = 4 \cdot 4 \cdot 4 = 64

Find the bottom.

So $4^{-3} = \dfrac{1}{64}$ .

-2^{-2} = -(2^{-2})

The base is 2. The minus sign stays out front.

-(2^{-2}) = -\frac{1}{2^2}

Rewrite the negative exponent.

-\frac{1}{2^2} = -\frac{1}{4}

Find the bottom.

7m^{-4} = 7 \cdot \frac{1}{m^4}

The exponent is on m only. Rewrite it.

\frac{7}{m^4}

Multiply.

\frac{3}{k^{-2}} = 3k^2

Move the factor on the bottom to the top and make the exponent positive.

\left(\frac{3}{5}\right)^{-2} = \left(\frac{5}{3}\right)^2

Flip the fraction and make the exponent positive.

\frac{5^2}{3^2} = \frac{25}{9}

Square the top and the bottom.

x^{-6} \cdot x^9 = x^{-6+9}

The bases match, so add the exponents.

x^3

Add.

\frac{20}{5} = 4

Divide the numbers.

\frac{p^3}{p^7} = p^{3-7} = p^{-4}

Subtract the exponents on p.

4p^{-4}q^{-2}

Put the parts together. The q stays the same.

4p^{-4}q^{-2} = \frac{4}{p^4q^2}

Move both negative powers to the bottom.

(3a^{-2}b)^2 = 3^2 \cdot (a^{-2})^2 \cdot b^2

Raise each factor to the 2nd power.

9a^{-4}b^2

Simplify. Multiply the exponents on a.

9a^{-4}b^2 \cdot a^5 = 9a^{-4+5}b^2

Multiply by $a^5$ . Add the exponents on a.

9a^1b^2 = 9ab^2

Add.

10.

5 \cdot 10^{-6} = 5 \cdot \frac{1}{1000000}

Rewrite the negative exponent as a fraction.

0.000005

Write it as a decimal. Move the decimal point 6 places left (see scientific notation)

5 \cdot 10^{-6} \cdot 10^3

For 1,000 grains, multiply by $10^3$ .

5 \cdot 10^{-6+3} = 5 \cdot 10^{-3}

Add the exponents.

5 \cdot 10^{-3} = 0.005

Write it as a decimal.

One grain is 0.000005 grams, and 1,000 grains are 0.005 grams.

Want more practice? The Algebra 1 Unit 6 Practice Pack has 49 pages of problems on this and the other lessons in the unit, with step-by-step answers and a cheat sheet.

FAQ

Why is any number to the zero power equal to 1?

Divide a power by itself. The quotient rule gives $\dfrac{x^4}{x^4} = x^0$ , and anything divided by itself is 1. So $x^0 = 1$ for any $x$ that is not 0.

What is 0 to the zero power?

$0^0$ is left undefined in algebra, because the patterns give two different answers. The rule $x^0 = 1$ is only for nonzero $x$ .

Does a negative exponent make a number negative?

No. A negative exponent makes a fraction. For example, $2^{-4} = \dfrac{1}{16}$ , which is positive.

How do you get rid of a negative exponent?

Move that factor to the other side of the fraction bar. Then make its exponent positive. Leave every other factor where it is.

Summary

Any nonzero number to the zero power is 1: $x^0 = 1$ .
A negative exponent means “one over”: $x^{-n} = \dfrac{1}{x^n}$ .
A negative exponent on the bottom moves to the top: $\dfrac{1}{x^{-n}} = x^n$ .
Only the factor attached to the exponent moves. Watch for parentheses.
A negative exponent never makes the value negative.

Previous lesson: Exponent Rules

Next lesson: Exponential Functions

Back to the Algebra 1 Index