In Exercises 39–64, rationalize each denominator.10-----³√5

Verified step by step guidance

1. Identify the problem: You need to rationalize the denominator of the expression \( \frac{10}{\sqrt[3]{5}} \).
2. To rationalize a cube root in the denominator, multiply both the numerator and the denominator by the square of the cube root. In this case, multiply by \( \sqrt[3]{5^2} = \sqrt[3]{25} \).
3. Write the expression as \( \frac{10}{\sqrt[3]{5}} \times \frac{\sqrt[3]{25}}{\sqrt[3]{25}} \).
4. Simplify the denominator: \( \sqrt[3]{5} \times \sqrt[3]{25} = \sqrt[3]{5^3} = 5 \).
5. The expression becomes \( \frac{10 \times \sqrt[3]{25}}{5} \). Simplify the fraction by dividing the numerator by 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 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, making the expression easier to work with and understand.

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 8 is 2, since 2 × 2 × 2 = 8. 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 involves using a specific factor that will eliminate the radical when multiplied. This method ensures that the overall value of the fraction remains unchanged while simplifying the expression.