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Evaluate powers with integer bases

A power with an integer base and a whole-number exponent represents repeated multiplication of the base, enabling exact evaluation while preserving the structure of the expression. Evaluation includes positive and negative bases, zero, and zero exponents for nonzero bases; the parity of the exponent determines whether a negative base produces a positive or negative result, and parentheses distinguish (−3)2(-3)^2 from −32-3^2. Negative and fractional exponents are not included.

Detailed Explanation: Evaluate powers with integer bases

A power tells you to multiply the base by itself the number of times shown by the exponent:

an=a⋅a⋅a⋯a⏟n factorsa^n=\underbrace{a\cdot a\cdot a\cdots a}_{n\text{ factors}}

Example: Evaluate (−3)4(-3)^4

  1. The base is −3-3, and the exponent is 44. This means multiply −3-3 by itself four times:

(−3)4=(−3)(−3)(−3)(−3)(-3)^4=(-3)(-3)(-3)(-3)

  1. Multiply in pairs:

(−3)(−3)=9and(−3)(−3)=9(-3)(-3)=9 \qquad \text{and} \qquad (-3)(-3)=9

  1. Multiply the results:

9⋅9=819\cdot 9=81

Therefore,

(−3)4=81\boxed{(-3)^4=81}

A negative base gives a positive result when the exponent is even, because the negative signs pair up. With an odd exponent, the result is negative; for example, (−3)3=−27(-3)^3=-27.

Remember that parentheses matter:

(−3)2=9(-3)^2=9

but

−32=−9-3^2=-9

The exponent applies to 33 before the negative sign. Also, any nonzero number raised to the zero power equals 11, such as 50=15^0=1, while 0n=00^n=0 for any positive whole-number exponent.

Learn by doing: Evaluate powers with integer bases

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