Negative Rational Exponents
Negative integral exponents were defined by using reciprocals, and so are negative
rational exponents. For example, 8^{2/3} is the reciprocal of 8^{2/3}.
So
Negative Rational Exponents
If m and n are positive integers, then
provided that a^{1/n} is defined and nonzero.
Three operations are involved in evaluating the expression a^{m/n}. The operations
(root, power, reciprocal) can be performed in any order, but the simplest way
to evaluate a^{m/n} is usually the following order.
Evaluating ^{m/n}
Example 2
Evaluating expressions with negative rational exponents
Evaluate each expression.
a) 4^{3/2 }
b) (27)^{1/3 }
c) (16)^{3/4 }
Solution
a) The square root of 4 is 2. The cube of 2 is 8. The reciprocal of 8 is
. So
b) The cube root of 27 is 3. The first power of 3 is 3. The reciprocal of
3 is
. So
c) The expression (16)^{3/4} is not a real number because it involves the fourth
root (an even root) of a negative number.
