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Rational Exponents

POWERS OF 1/2

Definition of b^1/2: (This is read as b to the one-half power.)  If the laws of exponents are to hold, then . Since the square of b^1/2 is b, b^1/2 is defined to be .

In general, raising a number to the ½ power is the same as taking the square root of the number.

OTHER FRACTIONAL POWERS

Definition of b^1/3: b^1/3 is defined to be , since its cube is b.

Definition of b^1/4: b^1/4 is defined to be , since (b^1/4)⁴ is b.

Definition of b^3/4:  Using the Power of a Power Law of Exponents in either of two ways:

Therefore, b^3/4 is defined to be either of the equivalent expressions .

The definition of any rational exponent is:

If p and q are integers, q ≠ 0 and b is a positive real number, then

.

Notice that in this case computing the root first is easier.  This is usually the case.

CUBE ROOTS AND OTHER RADICALS

Fractional exponents can also be written as radicals: