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.