Problem 50 - College Algebra

Question

Rationalize the denominator. $$\frac{3}{3+\sqrt{7}}$$

Step-by-Step Solution

Answer

The rationalized form of \(\frac{3}{3+\sqrt{7}}\) is \(\frac{9}{2}-\frac{3\sqrt{7}}{2}\).

Step 1: Analyze the problem

Given the expression \(\frac{3}{3+\sqrt{7}}\), we need to eliminate the square root from the denominator. A useful trick to accomplish this is to multiply the given fraction by \(\frac{3-\sqrt{7}}{3-\sqrt{7}}\), which is just a sophisticated form of 1.

Step 2: Multiply the given fraction

Now we multiply the given fraction \(\frac{3}{3+\sqrt{7}}\) by the clever form of 1 that we identified in Step 1, \(\frac{3-\sqrt{7}}{3-\sqrt{7}}\). This yields \(\frac{3(3-\sqrt{7})}{(3+\sqrt{7})(3-\sqrt{7})}\).

Step 3: Simplify the result

Proceed to multiply out the numerator and denominator: \(\frac{3(3-\sqrt{7})}{(3+\sqrt{7})(3-\sqrt{7})} = \frac{9-3\sqrt{7}}{3(3 - \sqrt{7}) +1(\sqrt{7} - 7)}) = \frac{9-3\sqrt{7}}{9-7} = \frac{9-3\sqrt{7}}{2}\).

Step 4: Simplify further

\(\frac{9-3\sqrt{7}}{2}\) can be simplified further by breaking it up into two separate fractions. Therefore, the final answer is \(\frac{9}{2}-\frac{3\sqrt{7}}{2}\).