Zero and Negative Exponents

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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.

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}.

The powers of 2 in a row: 2³ = 8, 2² = 4, 2¹ = 2, 2⁰ = 1, 2⁻¹ = 1/2 and 2⁻² = 1/4. An arrow labeled divide by 2 joins each box to the next one.
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.

RuleIn symbolsExample
Zero exponentx^0 = 19^0 = 1
Negative exponentx^{-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

  1. Find each factor with a zero exponent. Change it to 1.
  2. Find each factor with a negative exponent. Look at what the exponent is attached to.
  3. Move only that factor across the fraction bar, and make its exponent positive.
  4. Leave every other factor where it is, including numbers with no exponent.
  5. 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}.

(a) Any nonzero number to the zero power is 1: 6^0 = 1.

(b) The base is -4, inside the parentheses. So (-4)^0 = 1.

(c) There are no parentheses, so the base is just 4. The minus sign stays out front: -4^0 = -(4^0) = -1.

(d) Flip to the bottom and make the exponent positive: 3^{-2} = \dfrac{1}{3^2} = \dfrac{1}{9}.

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}.

(a) x^{-5} = \dfrac{1}{x^5}.

(b) The exponent is on y only. The 4 stays on top: 4y^{-2} = \dfrac{4}{y^2}.

(c) Now the exponent is on the whole group 4y. Move the group, then square both factors: (4y)^{-2} = \dfrac{1}{(4y)^2} = \dfrac{1}{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}}.

(a) The factor a^{-3} is on the bottom. Move it to the top and make the exponent positive: \dfrac{5}{a^{-3}} = 5a^3.

(b) Move m^{-2} down and n^{-4} up. Both exponents turn positive: \dfrac{m^{-2}}{n^{-4}} = \dfrac{n^4}{m^2}.

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}.

(a) Multiply the numbers: 2 \cdot 5 = 10. Add the exponents on each base: x^{-3+7} = x^4 and y^{4+(-6)} = y^{-2}. That gives 10x^4y^{-2}. Move y^{-2} to the bottom: \dfrac{10x^4}{y^2}.

(b) Divide the numbers: 12 \div 3 = 4. Subtract the exponents: a^{2-5} = a^{-3} and b^{-1-2} = b^{-3}. That gives 4a^{-3}b^{-3}, which is \dfrac{4}{a^3b^3}.

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?

Step 1: The length is 4 \cdot 10^{-3} meters. Since 10^{-3} = \dfrac{1}{1000}, that is \dfrac{4}{1000} = 0.004 meters.

Step 2: Divide the total length by one length: \dfrac{2}{4 \cdot 10^{-3}}. Split it up: \dfrac{2}{4} \cdot \dfrac{1}{10^{-3}} = \dfrac{1}{2} \cdot 10^3.

Step 3: \dfrac{1}{2} \cdot 1000 = 500. Check: 500 \cdot 0.004 = 2 meters.

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

Common Mistakes

MistakeExample of the errorFix
Making the number negative2^{-3} = -8A negative exponent means a fraction: 2^{-3} = \dfrac{1}{8}
Multiplying by the exponent2^{-3} = -6The exponent is not a factor: 2^{-3} = \dfrac{1}{8}
Moving the coefficient too3x^{-2} = \dfrac{1}{3x^2}Only x has the exponent: \dfrac{3}{x^2}
Saying the zero power is 08^0 = 08^0 = 1
Ignoring a minus sign out front-5^0 = 1The 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.

PowerFractionDecimal
10^2100100
10^11010
10^011
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

  1. Find the value of 15^0.
  2. Find the value of 4^{-3}.
  3. Find the value of -2^{-2}.
  4. Rewrite 7m^{-4} with positive exponents.
  5. Rewrite \dfrac{3}{k^{-2}} with positive exponents.
  6. Find the value of \left(\dfrac{3}{5}\right)^{-2}.
  7. Simplify x^{-6} \cdot x^{9}.
  8. Simplify \dfrac{20p^3q^{-2}}{5p^7}. Use positive exponents only.
  9. Simplify (3a^{-2}b)^2 \cdot a^5. Use positive exponents only.
  10. 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
  1. 15^0 = 1
  2. 4^{-3} = \dfrac{1}{4^3} = \dfrac{1}{64}
  3. The base is 2, and the minus sign stays out front: -2^{-2} = -\dfrac{1}{4}.
  4. Only m moves: \dfrac{7}{m^4}.
  5. Move k^{-2} to the top: 3k^2.
  6. Flip the fraction and make the exponent positive: \left(\dfrac{5}{3}\right)^2 = \dfrac{25}{9}.
  7. Add the exponents: x^{-6+9} = x^3.
  8. 20 \div 5 = 4 and p^{3-7} = p^{-4}. So 4p^{-4}q^{-2} = \dfrac{4}{p^4q^2}.
  9. (3a^{-2}b)^2 = 9a^{-4}b^2. Then 9a^{-4}b^2 \cdot a^5 = 9a^{1}b^2 = 9ab^2.
  10. 5 \cdot 10^{-6} = 0.000005 grams. For 1,000 grains: 5 \cdot 10^{-6} \cdot 10^3 = 5 \cdot 10^{-3} = 0.005 grams.

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

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