Negative Exponent
A negative exponent gives the reciprocal.
Formula
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Example
2^{-3}=1/8
How it works
A negative exponent tells you to take the reciprocal of the positive-power version of the same base. This works because exponent rules are built to make division by the same base undo repeated multiplication: for example, a^3 divided by a^5 must be 1/a^2, so the missing exponent has to behave like a negative power. In general, moving from a^n to a^-n means "flip it over" and keep the exponent positive.
When to use it
Use this when simplifying expressions with powers, especially when a base appears in the denominator. It is also common in algebra, scientific notation, and formulas involving fractions or inverse quantities.
Worked example, step by step
Start with 5^-2 Rewrite using the negative exponent rule: 5^-2 = 1 / 5^2 Evaluate the power: 5^2 = 25 So, 5^-2 = 1/25
Common mistakes to avoid
- keeping the negative sign in the denominator instead of flipping to a reciprocal
- thinking a^-n means -(a^n) instead of 1/(a^n)
- forgetting to make the exponent positive after taking the reciprocal
Did You Know?
A negative exponent means take the reciprocal, so 10⁻³ is one thousandth — the notation used across all of science.
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