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6.4 Rational Exponents
Rational Exponent: Another way to write a radical expression. Like the radical form, the
exponent form always indicates the principal root. The word exponent comes from the Latin
word meaning “place outside.”
Examples:
 √25 = 251/2
 ³√27 = 271/3
 4√16 = 161/4
Example 1: Simplifying Expressions with Rational Exponents
1a) 1251/3
Step 1: Rewrite as a Radical
³√125
Step 2: Simplify
=5
1/2 1/2
1b) 5 (5 ) =
√5 * √5
=5
1c) 101/3 (1001/3) =
1d) 161/4
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1e) 21/2 (81/2)
A rational exponent may have a numerator other than 1.
The property (am)n = amn shows how to rewrite an expression with an exponent that is
an improper fraction.
There are two ways to rewrite the expression.
Example: 1st) 253/2 = 25(³ * ½) = (25³)1/2 = √25³ or
2nd) 253/2= 25(½ * ³) = (251/2)³ = (√25)³
Property of Rational Exponents:
If the nth root of a is a real number and m is an integer, then
a1/n = n√a and am/n = n√am = n(√a)m. If m is negative, a ≠ 0.
Example 2: Converting to and From Radical Form
2: Write the exponential expressions in radical form.
2a) x3/5
2b) y-2.5
2c) y-3/8
2: Write the radical expressions in exponential form.
2e) √a³
2f) (5√b)²
2g) ³√x²
2h) (√y)³
2d) z.4
Example 3: Simplifying Numbers with Rational Exponents
 You can simplify a number with a rational exponent by using the properties of exponents
or by converting the expression to a radical expression.
3a) (-32) 3/5
3b) 4-3.5
3c)(-32) 4/5
Example 4: Writing Expressions in Simplest Form
 To write an expression with rational exponents in simplest form, write every exponent as
a positive number.
4a) (16y-8)-3/4
4b) (8x-15) -1/3