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Roots

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We say that such and such an exponent means that the base to which it is affixed is multiplied by itself the number of times specified by the exponent. We can use such an expression to determine the root of another number. Suppose 2^8 = 256. We would say that 2 is the 8th root of 256, that 4 is its 7th root, and so on. Keep in mind that 256 has two different 8th roots, because -2^8 and 2^8 both equal 256. All positive numbers have two square roots: its positive square root, and the negative counterpart of that positive square root. Indeed, positive numbers have square roots, 4th roots, 6th roots, 8th roots, and so on. These even roots of positive numbers are known as "principal" roots.

This is not true, however, of negative numbers. Negative numbers have no square roots. The reason for this is obvious: There is no number which, if multiplied by itself, yields a negative number. This is true of both negative and positive numbers multiplied by themselves. Negative numbers do, however, have real 3rd roots, real 5th roots, real 7th roots, real 9th roots, and so on. Rather than having two of such roots as is the case with the even roots of positive numbers, however, negative numbers, though they have odd roots, only ever have one root. Each root, moreover, is a negative number.

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