In Exercises 39–64, rationalize each denominator.3xy²-----------⁵√8xy³

Verified step by step guidance

1. Identify the denominator: \( \sqrt[5]{8xy^3} \).
2. To rationalize the denominator, multiply both the numerator and the denominator by \( \sqrt[5]{(8xy^3)^4} \) to make the denominator a perfect fifth power.
3. The expression becomes: \( \frac{3xy^2 \cdot \sqrt[5]{(8xy^3)^4}}{\sqrt[5]{8xy^3} \cdot \sqrt[5]{(8xy^3)^4}} \).
4. Simplify the denominator: \( \sqrt[5]{(8xy^3)^5} = 8xy^3 \).
5. Simplify the entire expression by multiplying and simplifying the numerator and the denominator.

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator involves rewriting a fraction so that the denominator is a rational number. This is often done by multiplying both the numerator and the denominator by a suitable expression that eliminates any roots or irrational numbers in the denominator. The goal is to simplify the expression while maintaining its value.

Radical Expressions

Radical expressions contain roots, such as square roots or cube roots. In this context, the expression involves a fifth root, which can complicate the process of rationalization. Understanding how to manipulate and simplify radical expressions is essential for effectively rationalizing denominators.

Properties of Exponents

Properties of exponents govern how to simplify expressions involving powers and roots. For instance, the property that states a^(m/n) = n√(a^m) helps in rewriting roots as fractional exponents. This understanding is crucial when dealing with expressions that include both variables and roots, as it aids in the simplification process.