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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 ( b 1 2 ) = b ( 1 2 ) 2 = b 1 = b . Since the square of b 1 2 is b , b 1 2 is定义为 b .

Example 1:

Simplify. 9 1 2

9 1 2 = 9 = 3

In general, raising a number to the 1 2 power is the same as taking thesquare rootof the number.

OTHER FRACTIONAL POWERS

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

Definition of b 1 4 : b 1 4 is defined to be b 4 , 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:

b 3 4 = ( b 1 4 ) 3 = ( b 4 ) 3 or b 3 4 = ( b 3 ) 1 4 = b 3 4

Therefore, b 3 4 is defined to be either of the equivalent expressions ( b 4 ) 3 or b 3 4 .

The definition of any rational exponent is:

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

b p / q = ( b q ) p = b p q .

Example 2:

Simplify. 27 2 3

27 2 3 = ( 27 3 ) 2 = 3 2 = 9

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:

x 3 = x 1 3

x 5 = x 1 5

x 2 5 = x 2 5