Algebra 101 - Unit 9

Lesson 55: Rational Exponents

Rational exponents provide an alternative way to write roots. A rational exponent is a fraction: the numerator is the power, and the denominator is the root. For example, x^(m/n) = n√(x^m) = (n√x)^m.

Examples

Example 1: x^(1/2) = √x

Example 2: x^(1/3) = ∛x

Example 3: x^(3/2) = (√x)^3 = √(x^3)

Example 4: 8^(2/3) = (∛8)^2 = 2^2 = 4

Example 5: 16^(3/4) = (√[4]{16})^3 = 2^3 = 8

Properties of Rational Exponents

• x^(a) × x^(b) = x^(a+b)
• x^(a) ÷ x^(b) = x^(a−b)
• (x^a)^b = x^(a*b)
• (xy)^a = x^a * y^a
• (x/y)^a = x^a / y^a

Practice Problems

1. Simplify: 27^(1/3)
2. Simplify: 32^(2/5)
3. Write as a radical: x^(5/2)
4. Write as a rational exponent: √[3]{x^4}
5. Simplify: 16^(3/4)