Problem 53 - College Algebra

Question

Rationalize the denominator. $$\frac{6}{\sqrt{5}+\sqrt{3}}$$

Step-by-Step Solution

Answer

The rationalized form of the given expression \(\frac{6}{\sqrt{5}+\sqrt{3}}\) is \(3(\sqrt{5} - \sqrt{3})\).

Step 1: Identify the conjugate of the denominator

The conjugate of a binomial expression of the form \(a + b\) is \(a - b\). So, the conjugate of \(\sqrt{5} + \sqrt{3}\) is \(\sqrt{5} - \sqrt{3}\).

Step 2: Rationalize by multiplying and dividing by the conjugate

Multiplying and dividing by the conjugate won't change the value of the original expression. Therefore, multiply both the numerator and the denominator by \(\sqrt{5} - \sqrt{3}\). So, \(\frac{6}{\sqrt{5} + \sqrt{3}}\) becomes \(\frac{6(\sqrt{5} - \sqrt{3})}{(\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3})}\).

Step 3: Simplify

Distribute 6 in the numerator 6(\(\sqrt{5}\) - \(\sqrt{3}\)) to get 6\(\sqrt{5}\) - 6\(\sqrt{3}\) and use the difference of squares formula \((a+b)(a-b) = a^2 - b^2\) in the denominator to get 5 - 3. The expression now is \(\frac{6\sqrt{5} - 6\sqrt{3}}{2}\).

Step 4: Simplify the expression

\(\frac{6\sqrt{5} - 6\sqrt{3}}{2}\) simplifies to \(3(\sqrt{5} - \sqrt{3})\) by factoring out 3 from the numerator.