In Exercises 39–64, rationalize each denominator. 3 ³√ ---- 4

Verified step by step guidance

1. Identify the expression: \( \frac{3}{\sqrt[3]{4}} \). 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 square of the cube root. In this case, multiply by \( \sqrt[3]{16} \) because \( \sqrt[3]{4} \times \sqrt[3]{16} = \sqrt[3]{64} = 4 \).
3. Write the expression as: \( \frac{3 \times \sqrt[3]{16}}{\sqrt[3]{4} \times \sqrt[3]{16}} \).
4. Simplify the denominator: \( \sqrt[3]{4} \times \sqrt[3]{16} = \sqrt[3]{64} = 4 \).
5. The expression is now \( \frac{3 \times \sqrt[3]{16}}{4} \), which has a rationalized 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 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 a Conjugate

Multiplying by a conjugate is a technique used to eliminate radicals from denominators. The conjugate of a binomial expression is formed by changing the sign between the two terms. For example, the conjugate of (a + b) is (a - b). This method is particularly useful when dealing with square roots, but can also be adapted for cube roots in certain cases.