Example 1: converting a root to a rational exponent

1. Write ∛(x⁵) as a rational exponent, then evaluate 8^2/3.
2. ∛(x⁵) = x^5/3. Also 8^2/3 = (∛8)² = 2² = 4.

Example 2: a negative rational exponent

• Evaluate 16^−3/4.
• Solution 16^−3/4 = 1 / 16^3/4. First find the fourth root: ⁴√16 = 2. Then raise it to the power 3: 2³ = 8. So the value is 1/8.

Example 3: a rational exponent greater than one

• Evaluate 27^4/3.
• Solution The cube root of 27 is 3. Raising 3 to the power 4 gives 3⁴ = 81.

Example 4: simplifying an expression

• Simplify (x^1/2 × x^3/2) ÷ x, where x > 0.
• Solution Adding the exponents in the product gives x^{1/2 + 3/2} = x². Dividing by x gives x^{2 − 1} = x.

Example 5: multiplying two different roots

• Write √(a³) × ∛(a²) as a single power of a, where a > 0.
• Solution √(a³) = a^3/2 and ∛(a²) = a^2/3. Adding the exponents: 3/2 + 2/3 = 9/6 + 4/6 = 13/6. The product is a^13/6.

Example 6: an equation with a rational exponent

Example 7: a challenging evaluation

• Evaluate 32^−2/5.
• Solution 32^−2/5 = 1 / 32^2/5. The fifth root of 32 is 2, and raising 2 to the power 2 gives 4. So the value is 1/4.