Rationalize each denominator. Assume all variables represent nonnegative numbers and that no denominators are 0. (1 + √3) / (3√5 + 2√3)

Verified step by step guidance

1. Identify the expression to rationalize: \(\frac{1 + \sqrt{3}}{3\sqrt{5} + 2\sqrt{3}}\).
2. To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(3\sqrt{5} + 2\sqrt{3}\) is \(3\sqrt{5} - 2\sqrt{3}\).
3. Multiply numerator and denominator by the conjugate: \(\frac{(1 + \sqrt{3})(3\sqrt{5} - 2\sqrt{3})}{(3\sqrt{5} + 2\sqrt{3})(3\sqrt{5} - 2\sqrt{3})}\).
4. Use the difference of squares formula for the denominator: \((a + b)(a - b) = a^2 - b^2\), where \(a = 3\sqrt{5}\) and \(b = 2\sqrt{3}\). Calculate \(a^2\) and \(b^2\) separately.
5. Expand the numerator by distributing each term in \((1 + \sqrt{3})\) with each term in \((3\sqrt{5} - 2\sqrt{3})\), then simplify the expression.

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any radicals (square roots) from the denominator of a fraction. This is done by multiplying the numerator and denominator by a suitable expression that will remove the radical, often the conjugate when dealing with binomials.

Conjugates of Binomials

The conjugate of a binomial expression a + b is a - b. Multiplying a binomial by its conjugate results in a difference of squares, which eliminates the square roots. This technique is essential for simplifying expressions with radicals in the denominator.

Properties of Square Roots and Nonnegative Variables

Square roots represent nonnegative values, and when variables are nonnegative, it simplifies handling radicals. Understanding that √a * √b = √(ab) and that variables under radicals are nonnegative helps avoid extraneous solutions and ensures correct simplification.