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Distinguish a negative base from a negative power value

A negative base is the quantity inside parentheses, as in (−3)n(-3)^n, whereas a negative power value is a result whose sign is negative, as in −32=−(32)=−9-3^2=-(3^2)=-9. Parentheses determine whether the negative sign is part of the base: a negative base raised to an even whole-number exponent produces a positive value, while an odd exponent produces a negative value; negative and fractional exponents are not included.

Detailed Explanation: Distinguish a negative base from a negative power value

A negative base has the negative sign inside parentheses:

  • In (−3)2(-3)^2, the base is −3-3.
  • The exponent 22 means multiply the base by itself:
(−3)2=(−3)(−3)=9(-3)^2=(-3)(-3)=9

A negative power value has the negative sign outside the power:

  • In −32-3^2, the base is 33, not −3-3.
  • Find the power first, then apply the negative sign:
−32=−(32)=−(9)=−9-3^2=-(3^2)=-(9)=-9

So, for the example,

(−3)2=9but−32=−9(-3)^2=9 \qquad\text{but}\qquad -3^2=-9

The parentheses decide whether the negative sign is part of the base. A negative base raised to an even exponent gives a positive result, while a negative base raised to an odd exponent gives a negative result.

Learn by doing: Distinguish a negative base from a negative power value

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