Section 5.2: Rational Exponents

This section covers rational exponents, their relationship to radicals, and how to simplify and solve expressions with fractional exponents.

Example 1: Converting to Radical Form

Express \( 27^{2/3} \) as a radical.

Step 1: Rewrite the exponent: \( 27^{2/3} = (\sqrt[3]{27})^2 \)

Step 2: Cube root of 27 is 3 → \( 3^2 = 9 \)

Simplify \( 16^{3/4} \).

Step 1: Rewrite as a radical: \( 16^{3/4} = (\sqrt[4]{16})^3 \)

Step 2: Fourth root of 16 is 2 → \( 2^3 = 8 \)

Express \( 64^{2/3} \) as a radical and simplify
Simplify \( 32^{3/5} \)
Write \( 81^{3/4} \) in radical form
Solve \( x^{3/2} = 27 \) for x
Simplify \( (8^{2/3}) \cdot (4^{1/2}) \)