where there are a total of n factors of b, and n can be any natural number, 1, 2, 3, …. The number b is called the base, n is called the exponent, and we say that we are “raising b to the n th power” (except when n is 2 we say that we are “squaring b” and when n is 3 we say that we are “cubing b”). Here are some examples showing exponentials and what they mean:
Multiplication property ![]()
Division property ![]()
Exponentiation property ![]()
On the other hand the numerator and denominator are equal:
Putting this together we find that b 0 = 1. In other words any base raised to the 0th power equals 1. |
On the other hand the numerator equals 1:
Putting this together we find that:
In other words any base raised to a negative power is the reciprocal of the same base raised to the corresponding positive power.
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Here is the proof:
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Here is the proof:
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Algebra Coach Exercises |