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Skill: Interpret negative integer exponents

Explanation and Free Practice Resources

Negative integer exponents indicate reciprocals: for any nonzero number bb and positive integer nn, b−n=1bnb^{-n}=\frac{1}{b^n}, so the negative sign changes the reciprocal relationship, not the sign of the value. This meaning connects exponent laws with powers of ten and scientific notation, such as 10−3=0.00110^{-3}=0.001; the scope is limited to nonzero numerical bases and integer exponents, excluding zero bases, fractional or irrational exponents, and more advanced generalized exponent theory.

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Detailed Explanation: Interpret negative integer exponents

A negative integer exponent means take the reciprocal. It does not mean the answer is negative.

For any nonzero base bb and positive integer nn,

b−n=1bn.b^{-n}=\frac{1}{b^n}.

Example: Evaluate 2−32^{-3}

  1. The exponent is negative, so rewrite the expression as a reciprocal:

2−3=1232^{-3}=\frac{1}{2^3}

  1. Evaluate the positive power:

23=2⋅2⋅2=82^3=2\cdot2\cdot2=8

  1. Write the result:

2−3=18\boxed{2^{-3}=\frac{1}{8}}

The negative sign in the exponent changed 232^3 into its reciprocal, 123\frac{1}{2^3}. It did not make the answer negative. This same idea explains why 10−3=1103=0.00110^{-3}=\frac{1}{10^3}=0.001.

Learn by doing: Interpret negative integer exponents

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