Why are there negative numbers in exponents?

Expert Answers

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Exponentiation is a process also referred to as repeated multiplication. If you have a positive number n, and a real number a, 

`a^n` means `a times a times ... times a`, done n times.

When an exponent is negative, as in `a^-n` ,we first take the reciprocal before performing the multiplication:

`a^-n = 1/a^n = 1/a times 1/a times ... times 1/a` .

In short, there can be negative numbers in exponents. If the exponent is negative, this just means that you have to take the reciprocal and apply the exponent to the reciprocal.

Let's have another example:

`5^-3 = (1/5)^3 = 1/5^3 = 1/125`

Notice how the negative exponent became positive when we got the reciprocal of the base.

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