Question

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

Accepted Answer

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}$$. To eliminate the cube roots in the denominator, multiply both numerator and denominator by the conjugate expression $$\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}$$. Write the new fraction as $$\frac{1}{\sqrt[3]{3} - \sqrt[3]{5}} 	imes \frac{\sqrt[3]{9} + \sqrt[3]{15} + \sqrt[3]{25}}{\sqrt[3]{9} + \sqrt[3]{15} + \sqrt[3]{25}}$$. Use the difference of cubes formula in the denominator: $$(\sqrt[3]{3} - \sqrt[3]{5})(\sqrt[3]{9} + \sqrt[3]{15} + \sqrt[3]{25}) = 3 - 5 = -2$$, which is a rational number. Express the fraction with the rationalized denominator as $$\frac{\sqrt[3]{9} + \sqrt[3]{15} + \sqrt[3]{25}}{-2}$$.

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

Verified video answer for a similar problem:

Key Concepts

Rationalizing the Denominator

Difference of Cube Roots and Conjugates

Properties of Cube Roots and Exponents

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