In Exercises 39–64, rationalize each denominator.3³√ -------xy²

Verified step by step guidance

1. Identify the expression: \( \frac{3}{\sqrt[3]{xy^2}} \). The goal is to rationalize the denominator.
2. To rationalize a cube root in the denominator, multiply both the numerator and the denominator by the expression that will make the denominator a perfect cube. In this case, multiply by \( \sqrt[3]{x^2y} \).
3. This gives: \( \frac{3 \cdot \sqrt[3]{x^2y}}{\sqrt[3]{xy^2} \cdot \sqrt[3]{x^2y}} \).
4. Simplify the denominator: \( \sqrt[3]{xy^2} \cdot \sqrt[3]{x^2y} = \sqrt[3]{(xy^2)(x^2y)} = \sqrt[3]{x^3y^3} = xy \).
5. The expression becomes: \( \frac{3 \cdot \sqrt[3]{x^2y}}{xy} \).

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator involves rewriting a fraction so that the denominator is a rational number. This is often necessary when the denominator contains a radical, such as a square root or cube root. The goal is to eliminate the radical from the denominator, which can simplify calculations and make the expression easier to work with.

Cube Roots

A cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 27 is 3, since 3 × 3 × 3 = 27. In the context of rationalizing denominators, understanding how to manipulate cube roots is essential for simplifying expressions that involve them.

Multiplying by the Conjugate

When rationalizing denominators that contain roots, one common technique is to multiply both the numerator and the denominator by a form of the conjugate. For cube roots, this may involve using a specific factor that will eliminate the radical when multiplied. This method is crucial for transforming the expression into a more manageable form.