Concept Check: By what number should the numerator and denominator of $\frac{1}{\text{∛}3-\text{∛}5}$ be multiplied in order to rationalize the denominator? Write this fraction with a rationalized denominator.

Verified step by step guidance

1. Identify the denominator as \(\sqrt[3]{3} - \sqrt[3]{5}\). To rationalize a denominator involving cube roots, we use the fact that \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\), where \(a = \sqrt[3]{3}\) and \(b = \sqrt[3]{5}\).
2. To eliminate the cube roots in the denominator, multiply both numerator and denominator by the conjugate expression \(\sqrt[3]{3^2} + \sqrt[3]{3} \cdot \sqrt[3]{5} + \sqrt[3]{5^2}\), which is \(\sqrt[3]{9} + \sqrt[3]{15} + \sqrt[3]{25}\).
3. Write the new fraction as \(\frac{1}{\sqrt[3]{3} - \sqrt[3]{5}} \times \frac{\sqrt[3]{9} + \sqrt[3]{15} + \sqrt[3]{25}}{\sqrt[3]{9} + \sqrt[3]{15} + \sqrt[3]{25}}\).
4. Use the difference of cubes formula in the denominator: \((\sqrt[3]{3} - \sqrt[3]{5})(\sqrt[3]{9} + \sqrt[3]{15} + \sqrt[3]{25}) = 3 - 5 = -2\), which is a rational number.
5. Express the fraction with the rationalized denominator as \(\frac{\sqrt[3]{9} + \sqrt[3]{15} + \sqrt[3]{25}}{-2}\).

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any roots or irrational numbers from the denominator of a fraction. This is done by multiplying the numerator and denominator by a suitable expression that makes the denominator a rational number, simplifying the expression.

Difference of Cube Roots and Conjugates

When dealing with expressions like ∛a - ∛b, the conjugate used to rationalize is based on the sum and product of cube roots, similar to the difference of cubes formula. Multiplying by the expression ∛a² + ∛a∛b + ∛b² helps eliminate cube roots in the denominator.

Properties of Cube Roots and Exponents

Understanding how cube roots and exponents interact is essential. For example, (∛x)³ = x, and multiplying cube roots follows the rule ∛a * ∛b = ∛(ab). These properties allow simplification when multiplying expressions involving cube roots.