Rationalize each denominator. Assume all variables represent nonnegative numbers and that no denominators are 0. 5√x / (2√x + √y)

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Identify the expression to rationalize: \(\frac{5\sqrt{x}}{2\sqrt{x} + \sqrt{y}}\).

Recognize that the denominator is a binomial involving square roots, so multiply numerator and denominator by the conjugate of the denominator to rationalize it. The conjugate of \(2\sqrt{x} + \sqrt{y}\) is \(2\sqrt{x} - \sqrt{y}\).

Multiply both numerator and denominator by the conjugate: \(\frac{5\sqrt{x}}{2\sqrt{x} + \sqrt{y}} \times \frac{2\sqrt{x} - \sqrt{y}}{2\sqrt{x} - \sqrt{y}}\).

Use the difference of squares formula for the denominator: \((a + b)(a - b) = a^2 - b^2\), where \(a = 2\sqrt{x}\) and \(b = \sqrt{y}\). Calculate \(a^2\) and \(b^2\) carefully.

Expand the numerator by distributing \(5\sqrt{x}\) over \(2\sqrt{x} - \sqrt{y}\), then write the simplified numerator over the simplified denominator to complete the rationalization.

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Rationalizing the Denominator

Rationalizing the denominator involves eliminating any radicals (square roots) from the denominator of a fraction. This is done to simplify the expression and make it easier to work with. Typically, this is achieved by multiplying the numerator and denominator by a conjugate or an appropriate radical.

Conjugates of Binomials

The conjugate of a binomial expression like (a + b) is (a - b). Multiplying a binomial by its conjugate uses the difference of squares formula, which removes the square roots in the denominator by turning them into a rational number. This technique is essential for rationalizing denominators with sums or differences of radicals.

Properties of Square Roots and Nonnegative Variables

Square roots represent nonnegative values, and when variables are nonnegative, expressions under the root are well-defined. Understanding that √x and √y are nonnegative helps avoid sign ambiguities during simplification. This assumption ensures the rationalization process is valid and the simplified form is correct.