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 Dependent Variable

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# 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 82/3. So

Negative Rational Exponents

If m and n are positive integers, then provided that a1/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.

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