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Negative Exponents

Negative Exponents

Negative exponents are a way of indicating reciprocals.

Rule:                 a to the power of negative n equals 1 divided by a to the power of n

Examples:         5 to the power of negative 2 equals 1 over 5 squared equals 1 over 25

   (2/3)^(-3) equals (3/2)^3 equals 27/8, showing a negative exponent becomes positive when the fraction is flipped.

 

 

See also

Fractions, rational exponents

Key Formula

a−n=1anwhere a≠0a^{-n} = \frac{1}{a^n} \quad \text{where } a \neq 0
Where:
  • aa = The base — any nonzero number
  • nn = A positive integer (or any positive number) used as the exponent

Worked Example

Problem: Simplify 5^{-3}.
Step 1: Apply the negative exponent rule: move the base to the denominator and make the exponent positive.
5−3=1535^{-3} = \frac{1}{5^3}
Step 2: Evaluate 535^3 by multiplying 5 by itself three times.
53=5×5×5=1255^3 = 5 \times 5 \times 5 = 125
Step 3: Write the final result.
5−3=11255^{-3} = \frac{1}{125}
Answer: 5−3=11255^{-3} = \dfrac{1}{125}

Another Example

Problem: Simplify the expression 2−43−2\dfrac{2^{-4}}{3^{-2}}.
Step 1: Rewrite each negative exponent as a reciprocal.
2−43−2=124132\frac{2^{-4}}{3^{-2}} = \frac{\frac{1}{2^4}}{\frac{1}{3^2}}
Step 2: Dividing by a fraction is the same as multiplying by its reciprocal.
124×321=3224\frac{1}{2^4} \times \frac{3^2}{1} = \frac{3^2}{2^4}
Step 3: Evaluate the powers and simplify.
916\frac{9}{16}
Answer: 2−43−2=916\dfrac{2^{-4}}{3^{-2}} = \dfrac{9}{16}

Frequently Asked Questions

Does a negative exponent make the result negative?
No. A negative exponent does not make the value negative. It creates a fraction (reciprocal). For instance, 2−3=182^{-3} = \frac{1}{8}, which is positive. A negative exponent only changes where the base sits — it moves it from numerator to denominator, or vice versa.
What happens when a negative exponent is in the denominator?
A negative exponent in the denominator moves the base up to the numerator with a positive exponent. For example, 1x−4=x4\frac{1}{x^{-4}} = x^4. You can think of it as applying the reciprocal rule twice: the base flips from denominator to numerator.

Negative exponent vs. Negative base

A negative exponent (2−3=182^{-3} = \frac{1}{8}) tells you to take a reciprocal — the result is a fraction, not a negative number. A negative base ((−2)3=−8(-2)^3 = -8) means the number being multiplied is negative, which can produce a negative result. Students often confuse the two, but the negative sign plays completely different roles in each case.

Why It Matters

Negative exponents appear constantly in science, especially in units like m⋅s−2\text{m} \cdot \text{s}^{-2} (meters per second squared) for acceleration. They also let you write very small numbers cleanly — scientific notation uses negative exponents to express quantities like 3.0×10−83.0 \times 10^{-8}. Understanding them is essential for simplifying algebraic expressions and working with exponential functions.

Common Mistakes

Mistake: Thinking a−na^{-n} gives a negative answer, such as writing 2−3=−82^{-3} = -8.
Correction: The negative exponent creates a reciprocal, not a negative value. 2−3=123=182^{-3} = \frac{1}{2^3} = \frac{1}{8}, which is positive.
Mistake: Applying the negative exponent to a coefficient that isn't part of the base, such as writing 3x−2=13x23x^{-2} = \frac{1}{3x^2}.
Correction: The exponent −2-2 applies only to xx, not to the coefficient 3. The correct simplification is 3x−2=3x23x^{-2} = \frac{3}{x^2}. Only the base directly attached to the exponent gets moved.

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