A power with a negative integer base represents repeated multiplication of that base; its value is positive when the exponent is even and negative when the exponent is odd, with exponent zero yielding 1. The distinction between and is essential because parentheses determine whether the negative sign is part of the base; this scope excludes negative or noninteger exponents.
A power with a negative integer base means repeated multiplication of the negative number. The parentheses show that the negative sign is part of the base.
For example, evaluate
The base is , and the exponent is . So multiply five times:
Pair the factors:
There are five negative factors, so one negative factor is left over. Therefore, the product is negative.
Multiply the absolute values:
Thus,
An even exponent gives a positive answer because the negative factors pair up. An odd exponent gives a negative answer. Also, any nonzero number raised to the zero power equals :
Be careful with parentheses:
because the base is , but
because the exponent applies only to , and the negative sign remains outside the power.
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