In Exercises 39–64, rationalize each denominator. 5 ³√ ----- y²

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, especially when they appear in the denominator of a fraction.

Multiplying by the Conjugate

Multiplying by the conjugate is a technique used to eliminate radicals from denominators. For expressions involving cube roots, this may involve multiplying both the numerator and denominator by a form of the radical that will result in a rational number. This method is particularly useful for simplifying expressions and ensuring that the final result is in a more manageable form.