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

Verified step by step guidance

1. Multiply the numerator and the denominator by the conjugate of the denominator, which is \(2\sqrt{x} - \sqrt{y}\).
2. The expression becomes \(\frac{5\sqrt{x} \cdot (2\sqrt{x} - \sqrt{y})}{(2\sqrt{x} + \sqrt{y})(2\sqrt{x} - \sqrt{y})}\).
3. Simplify the numerator: Distribute \(5\sqrt{x}\) to both terms in the conjugate, resulting in \(5\sqrt{x} \cdot 2\sqrt{x} - 5\sqrt{x} \cdot \sqrt{y}\).
4. Simplify the denominator using the difference of squares formula: \((2\sqrt{x})^2 - (\sqrt{y})^2\).
5. Simplify further by calculating \((2\sqrt{x})^2 = 4x\) and \((\sqrt{y})^2 = y\), so the denominator becomes \(4x - y\).

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any radical expressions from the denominator of a fraction. This is typically achieved by multiplying both the numerator and the denominator by a suitable expression that will result in a rational number in the denominator. For example, if the denominator is a binomial involving a square root, one would multiply by the conjugate of that binomial.

Conjugates

The conjugate of a binomial expression is formed by changing the sign between the two terms. For instance, the conjugate of (a + b) is (a - b). When multiplying a binomial by its conjugate, the result is a difference of squares, which eliminates the radical in the denominator. This technique is essential for rationalizing denominators that contain two terms.

Properties of Exponents and Radicals

Understanding the properties of exponents and radicals is crucial for manipulating expressions involving square roots. Key properties include the fact that √(a*b) = √a * √b and that (√a)^2 = a. These properties allow for simplification of expressions and are particularly useful when rationalizing denominators, as they help in rewriting and simplifying the resulting expressions.