Roots can also extend to a higher order than cube roots. The 4th root of a
number is a number that, when taken to the fourth power, is equal to the given
number. The 5th root of a number is a number that, when taken to the fifth
power, is equal to the given number, and so on. The 4th root is denoted by an
exponent of "1/4", the 5th root is denoted by an exponent of "1/5"; every root
is denoted by an exponent with 1 in the numerator and the order of root in the
denominator.

An odd root of a negative number is a negative number. We cannot take an even
root of a negative number. For example, (- 27)^{1/3} = - 3, but
(- 81)^{1/4} does not exist.

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Fractional Exponents

We have just learned that a fractional exponent with "1" in the numerator is a
root of some sort. But what would an exponent of "2/3" mean? Or an exponent of
"-5/2"?

In a fractional exponent, the numerator is the power to which the number
should be taken and the denominator is the root that should be taken. For
example, 64^{2/3} means "square 64 and take the cube root of the
result" or "take the cube root of 64 and square the result. This works out to
16.

A negative fractional exponent works just like a negative exponent. First,
we switch the numerator and the denominator of the base number, and then
we apply the positive exponent. For example, (9/25)^{-5/2} = (25/9)^{5/2} = (25^{5/2})/(9^{5/2}) = "the square root
of 25 to the fifth power over the square root of 9 to the fifth power"
= 3, 125/243. 27^{-1/3} = (1/27)^{1/3} = (1^{1/3})/(27^{1/3}) = 1/3.

Again, we cannot take a negative number to a fractional power if the denominator
of the exponent is even.