6-4 Rational Exponents

Objective

To simplify expressions with rational exponents

Image Long Description

If a x = a 3 4 , eh to the x , equals , the fourth , root of eh cubed end root , , comma  then by definition, a x ⋅ a x ⋅ a x ⋅ a x = a 3 . eh to the x , dot , eh to the x , dot , eh to the x , dot , eh to the x , equals , eh cubed , .  By adding exponents, a 4 x = a 3 , eh super 4 x end super , equals , eh cubed , comma  so x is 3 4 . 3 fourths , .  This suggests an alternative notation for radical expressions in which, for example, a 3 4 = a 3 4 . the fourth , root of eh cubed end root , , equals , eh super 3 fourths end super , .

Essential Understanding You can write a radical expression in an equivalent form using a fractional (rational) exponent instead of a radical sign.

In general, x n = x 1 n the th , root of x , , equals , x super 1 over n end super  for any positive integer n. Like the radical form, the exponent form indicates the principal root.

• 36 = 36 1 2 square root of 36 equals , 36 super and 1 half end super
• 64 3 = 64 1 3 cube root of 64 , , equals , 64 super and 1 third end super
• 16 4 = 16 1 4 the fourth , root of 16 , , equals , 16 super and 1 fourth end super

Table of Contents