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Distinguish negative powers from powers with negative bases

A negative exponent, such as 2−32^{-3}, indicates a reciprocal: 2−3=123=182^{-3}=\frac{1}{2^3}=\frac18; it does not make the base negative. A negative base, such as (−2)3(-2)^3 or (−2)4(-2)^4, remains signed, with the result negative for an odd exponent and positive for an even exponent; parentheses distinguish the base from a leading subtraction sign, as in (−2)2(-2)^2 versus −22-2^2. The scope is numerical powers with integer exponents, not generalized rational or real exponents.

Detailed Explanation: Distinguish negative powers from powers with negative bases

A negative exponent tells you to take a reciprocal. It does not make the base negative:

a−n=1ana^{-n}=\frac{1}{a^n}

Parentheses show whether the negative sign is part of the base.

Example: Evaluate each expression.

1. 2−32^{-3}

The base is positive 22, and the exponent is negative. Take the reciprocal:

2−3=123=182^{-3}=\frac{1}{2^3}=\frac18

So, 2−32^{-3} is positive.

2. (−2)3(-2)^3

The base is negative because the negative sign is inside the parentheses. The exponent is odd, so the answer is negative:

(−2)3=(−2)(−2)(−2)=−8(-2)^3=(-2)(-2)(-2)=-8

3. (−2)4(-2)^4

Again, the base is negative, but the exponent is even. A negative number multiplied an even number of times gives a positive result:

(−2)4=(−2)(−2)(−2)(−2)=16(-2)^4=(-2)(-2)(-2)(-2)=16

4. −22-2^2

There are no parentheses around −2-2, so the exponent applies only to 22. The subtraction sign is applied afterward:

−22=−(22)=−4-2^2=-(2^2)=-4

Remember:

  • A negative exponent means reciprocal: 2−3=182^{-3}=\frac18.
  • A negative base must be shown with parentheses: (−2)3(-2)^3.
  • For a negative base, an odd power is negative and an even power is positive.
  • (−2)2(-2)^2 and −22-2^2 are different: 44 versus −4-4.

Learn by doing: Distinguish negative powers from powers with negative bases

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    Exponents - Negative Exponents, Negative Base (to Fraction Exponent Form)

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