Powers, Exponents, and Roots

Square Roots

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.

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, 642/3 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 = (255/2)/(95/2) = "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 = (11/3)/(271/3) = 1/3.

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