Key takeaway: A negative exponent tells you to take the reciprocal of the base raised to the matching positive power, so \(a^{-n} = \frac{1}{a^{n}}\) for any nonzero \(a\).
Grade 88.EE.A.1Numbers and Operations
Definition of Negative Exponent
A negative exponent does not make a number negative. It flips the position of the power: whatever is raised to a negative exponent moves across the fraction bar and the exponent changes sign. So \(5^{-3}\) is the reciprocal of \(5^3\), which is \(\frac{1}{125}\).
The reason is a pattern. Each time the exponent drops by 1, the value is divided by the base: 100, 10, 1, then \(\frac{1}{10}\), \(\frac{1}{100}\). Continuing past zero gives the fractions. This is also what lets scientific notation describe very small numbers, since \(10^{-6}\) is one millionth.
Formula and notation
For a nonzero base \(a\) and a positive integer \(n\):
\[a^{-n} = \frac{1}{a^{n}} \qquad \text{and} \qquad \frac{1}{a^{-n}} = a^{n}\]
A negative exponent only attaches to the one factor it touches, so \(3x^{-2} = \frac{3}{x^{2}}\).
Key facts
- The sign of the exponent never changes the sign of the base.
- Moving a factor across the fraction bar flips the sign of its exponent.
- All other exponent laws still work when exponents are negative.
- The expression \(10^{-3}\) equals 0.001, which is why small numbers need it.
Negative Exponent example
Evaluate \(4^{-2} \cdot 4^{5}\). Then rewrite \(3x^{-2}\) and \(\dfrac{7}{y^{-4}}\) with positive exponents.
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Same base, so add exponents: \(4^{-2+5} = 4^{3} = 64\).
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Move \(x^{-2}\) below the bar: \(3x^{-2} = \dfrac{3}{x^{2}}\).
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Move \(y^{-4}\) above the bar: \(\dfrac{7}{y^{-4}} = 7y^{4}\).
The results are 64, \(\dfrac{3}{x^{2}}\) and \(7y^{4}\).
Common mistake
Watch out
Elias computes \(2^{-3} = -8\), thinking the minus sign in the exponent makes the number negative.
Why it is wrong
The exponent’s sign signals a reciprocal, not a negative value. \(2^{-3} = \frac{1}{2^{3}} = \frac{1}{8}\). A power of a positive base stays positive whatever the exponent. Think of halving a pizza again and again: the slices get tiny, but they never dip below zero.
Related terms and lessons
Browse the glossary: A–Z list · Grade 8 vocabulary · Numbers and Operations