Rationalize each denominator. See Example 8. 6/(√5 + √3)

Verified step by step guidance

1. Identify the expression to rationalize: \(\frac{6}{\sqrt{5} + \sqrt{3}}\).
2. Recall that to rationalize a denominator with two terms involving square roots, multiply numerator and denominator by the conjugate of the denominator. The conjugate of \(\sqrt{5} + \sqrt{3}\) is \(\sqrt{5} - \sqrt{3}\).
3. Multiply both numerator and denominator by the conjugate: \(\frac{6}{\sqrt{5} + \sqrt{3}} \times \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} - \sqrt{3}}\).
4. Use the difference of squares formula for the denominator: \((a + b)(a - b) = a^2 - b^2\). Here, \(a = \sqrt{5}\) and \(b = \sqrt{3}\), so the denominator becomes \$5 - 3$.
5. Simplify the numerator by distributing 6: \(6(\sqrt{5} - \sqrt{3})\), and simplify the denominator to \(2\). The expression is now rationalized.

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any square roots or irrational numbers from the denominator of a fraction. This is done to simplify the expression and make it easier to work with, often by multiplying numerator and denominator by a conjugate or an appropriate radical.

Conjugates of Binomials

The conjugate of a binomial expression like (√5 + √3) is (√5 - √3). Multiplying a binomial by its conjugate results in a difference of squares, which removes the square roots in the denominator, producing a rational number.

Difference of Squares Formula

The difference of squares formula states that (a + b)(a - b) = a² - b². This property is used to simplify expressions involving conjugates, especially when rationalizing denominators containing sums or differences of square roots.