Negative Exponents

Learn what a negative exponent means, why it gives a reciprocal and not a negative number, and how to rewrite any expression with positive exponents.

Intermediate25 min18 problems
Practice Problems

Definition

A negative exponent tells you to take the reciprocal of the power with the same positive exponent:
a−n=1an(a≠0)a^{-n} = \frac{1}{a^n} \qquad (a \neq 0)
For example, 2−3=123=182^{-3} = \frac{1}{2^3} = \frac{1}{8}.
Where does this come from? Watch the powers of 2. Every time the exponent goes down by 1, you divide by 2:
23=8,  22=4,  21=2,  20=1,  2−1=12,  2−2=14,  2−3=182^3 = 8,\; 2^2 = 4,\; 2^1 = 2,\; 2^0 = 1,\; 2^{-1} = \frac{1}{2},\; 2^{-2} = \frac{1}{4},\; 2^{-3} = \frac{1}{8}
The pattern simply keeps going below the zero exponent.
Key facts:
  • A negative exponent does not make the number negative: 2−3=182^{-3} = \frac{1}{8} is positive.
  • It works in both directions: 1a−n=an\frac{1}{a^{-n}} = a^n, so a factor can move across the fraction bar if its exponent changes sign.
  • A fraction to a negative power flips: (ab)−n=(ba)n\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n.
  • Powers of 10 with negative exponents are small decimals: 10−3=11000=0.00110^{-3} = \frac{1}{1000} = 0.001.

Try it now

What is the value of 2−32^{-3}?

Worked Examples

Evaluate 2−32^{-3}.
1

Apply the negative exponent rule

a−n=1ana^{-n} = \frac{1}{a^n} → 2−3=1232^{-3} = \frac{1}{2^3}
2

Calculate the power in the denominator

23=2×2×2=82^3 = 2 \times 2 \times 2 = 8 → 18\frac{1}{8}
3

Check the sign

The base 22 is positive, so the result is positive → 18=0.125\frac{1}{8} = 0.125

Common Mistakes

Thinking a negative exponent makes the number negative: 2−3=−82^{-3} = -8

Why it's wrong: The minus sign is in the exponent, not in front of the number. It means "reciprocal", not "negative".

Correct: 2−3=123=182^{-3} = \frac{1}{2^3} = \frac{1}{8}, a positive number.

Moving the coefficient too: 2x−3=12x32x^{-3} = \frac{1}{2x^3}

Why it's wrong: The exponent −3-3 belongs only to xx. The 22 has exponent 11 and stays where it is.

Correct: 2x−3=2x32x^{-3} = \frac{2}{x^3}. Only write 1(2x)3\frac{1}{(2x)^3} when the problem says (2x)−3(2x)^{-3} with parentheses.

Flipping a fraction but forgetting the power, or making it negative: (23)−2=32\left(\frac{2}{3}\right)^{-2} = \frac{3}{2} or −49-\frac{4}{9}

Why it's wrong: The minus sign flips the fraction, and the 22 still squares it. Both parts of the exponent have a job.

Correct: (23)−2=(32)2=94\left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^{2} = \frac{9}{4}

Subtracting a negative exponent the wrong way: x5x−2=x3\frac{x^5}{x^{-2}} = x^{3}

Why it's wrong: The quotient rule subtracts the whole exponent, including its sign: 5−(−2)5 - (-2), not 5−25 - 2.

Correct: x5x−2=x5−(−2)=x7\frac{x^5}{x^{-2}} = x^{5-(-2)} = x^{7}

Interactive Visual

Exponents as repeated factors

Quotient rule

Same base, divide: subtract the exponents.

x2x5\frac{x^{2}}{x^{5}}

Write each power as repeated factors

xx·xx
xx·xx·xx·xx·xx

2 common factors cancel, top and bottom

3 factors are left at the bottom: a negative exponent

1x3=x−3\displaystyle \frac{1}{x^{3}} = x^{-3}

Result

x2x5\displaystyle \frac{x^{2}}{x^{5}}==x2−5\displaystyle x^{2 - 5}==x−3\displaystyle x^{-3}==1x3\displaystyle \frac{1}{x^{3}}

This works for every base except 0

10³ = 1000
3

See how powers of a number grow on the number line. Change the base and exponent.

2³ = 8
3

See how powers of a number grow on the number line. Change the base and exponent.

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Practice Problems

18 problems
Problem 1 of 18
Easy
What is the value of 2−32^{-3}?

Why It Matters

Negative exponents let us write very small numbers as easily as very large ones, and they make the exponent rules work in every case:
  • Science: a red blood cell is about 8×10−68 \times 10^{-6} meters wide, and a nanosecond is 10−910^{-9} seconds
  • Metric prefixes: milli means 10−310^{-3} and micro means 10−610^{-6}, so 1 mm =10−3= 10^{-3} m
  • Photography: every stop on a camera halves the light, so 3 stops let in 2−3=182^{-3} = \frac{1}{8} of the light
  • Algebra: with the quotient rule, x2x5=x2−5=x−3\frac{x^2}{x^5} = x^{2-5} = x^{-3}. Without negative exponents, this answer would have no meaning
When you can switch between x−3x^{-3} and 1x3\frac{1}{x^3} without thinking, simplifying long expressions becomes routine.

Real World Applications

Tiny Measurements in Science

Scientists write very small lengths with negative powers of 10. The metric prefixes milli, micro and nano stand for $10^{-3}$, $10^{-6}$ and $10^{-9}$.

Example:

A millimeter is 10−310^{-3} m =0.001= 0.001 m, and a typical bacterium is about 2×10−62 \times 10^{-6} m long.

1Try It Yourself

A human hair is about 10−410^{-4} meters thick.

Write the thickness of the hair as a decimal number of meters.

Step 1: Write the mathematical expression

Rewrite 10−410^{-4} as 1104\frac{1}{10^4} and divide:

Camera Exposure Stops

Photographers measure light in stops. Each stop darker halves the light, so $n$ stops let in $2^{-n}$ of the original light.

Example:

A filter that is 3 stops dark lets in 2−3=182^{-3} = \frac{1}{8} of the light.

2Try It Yourself

A photographer puts a 5 stop filter in front of the lens.

What fraction of the light reaches the camera?

Step 1: Write the mathematical expression

Each stop halves the light, so the fraction is 2−52^{-5}:

Diluting Samples in a Lab

In a lab, a sample is often diluted 1 to 10 several times. After $n$ such dilutions, the concentration is $10^{-n}$ times the original.

Example:

After 3 dilutions, a sample has 10−3=0.00110^{-3} = 0.001 of its original concentration.

3Try It Yourself

A sample has 20000 bacteria per milliliter. It is diluted 1 to 10 four times in a row.

How many bacteria per milliliter are left?

Step 1: Write the mathematical expression

Multiply the starting amount by 10−410^{-4}:

Key Takeaways

  • 1A negative exponent means reciprocal: a−n=1ana^{-n} = \frac{1}{a^n} for a≠0a \neq 0
  • 2A negative exponent does not make the number negative: 2−3=182^{-3} = \frac{1}{8}
  • 3The rule works both ways: 1a−n=an\frac{1}{a^{-n}} = a^n, so a factor crosses the fraction bar by changing the sign of its exponent
  • 4A fraction to a negative power flips: (ab)−n=(ba)n\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n
  • 5Negative powers of 10 are small decimals: 10−3=0.00110^{-3} = 0.001
  • 6The product, quotient and power of a power rules all work with negative exponents; finish by rewriting with positive exponents

Frequently Asked Questions

By the product rule, a1×a−1=a0=1a^{1} \times a^{-1} = a^{0} = 1. The number you multiply by aa to get 11 is 1a\frac{1}{a}, so a−1=1aa^{-1} = \frac{1}{a}.
By the product rule, a1×a−1=a0=1a^{1} \times a^{-1} = a^{0} = 1. The number you multiply by aa to get 11 is 1a\frac{1}{a}, so a−1=1aa^{-1} = \frac{1}{a}.
Only if the base itself is negative. For example, (−2)−3=1(−2)3=−18(-2)^{-3} = \frac{1}{(-2)^3} = -\frac{1}{8}. The negative exponent never adds a minus sign by itself.
0−20^{-2} would mean 102=10\frac{1}{0^2} = \frac{1}{0}, and division by zero is not defined. That is why the rule says a≠0a \neq 0.
x−3x^{-3} and 1x3\frac{1}{x^3} are equal, so neither is wrong. Many teachers and textbooks ask for positive exponents in the final answer, so rewrite at the end unless the task says otherwise.

Glossary

Negative exponent
An exponent less than zero. a−na^{-n} means 1an\frac{1}{a^n}.
Reciprocal
The number you multiply by to get 1. The reciprocal of 55 is 15\frac{1}{5}, and the reciprocal of 23\frac{2}{3} is 32\frac{3}{2}.
Base
The number being multiplied. In 10−310^{-3}, the base is 1010.
Positive exponent form
An expression rewritten so that every exponent is positive, for example y2x3\frac{y^2}{x^3} instead of x−3y2x^{-3}y^{2}.
Coefficient
The number in front of a variable. In 2x−32x^{-3}, the coefficient is 22 and it is not affected by the exponent.

Formula Card

Negative exponent

a−n=1ana^{-n} = \frac{1}{a^n}

A negative exponent means reciprocal ($a \neq 0$)

Negative exponent in the denominator

1a−n=an\frac{1}{a^{-n}} = a^n

A factor moves across the fraction bar and its exponent changes sign

Fraction to a negative power

(ab)−n=(ba)n\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n

Flip the fraction, then use the positive exponent

Small powers of 10

10−n=0.0…0⏟n−1110^{-n} = 0.\underbrace{0\ldots0}_{n-1}1

For example $10^{-3} = 0.001$

Today's challengeOne problem, three tries

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