/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 50 Rationalize the denominator. $... [FREE SOLUTION] | 91影视

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Chapter 0: Problem 50

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

Short Answer

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

Step by step solution

01

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.
02

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 \(\frac{3(3-\sqrt{7})}{(3+\sqrt{7})(3-\sqrt{7})}\).
03

Simplify the result

Proceed to multiply out the numerator and denominator: \(\frac{3(3-\sqrt{7})}{(3+\sqrt{7})(3-\sqrt{7})} = \frac{9-3\sqrt{7}}{3(3 - \sqrt{7}) +1(\sqrt{7} - 7)}) = \frac{9-3\sqrt{7}}{9-7} = \frac{9-3\sqrt{7}}{2}\).
04

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 \(\frac{9}{2}-\frac{3\sqrt{7}}{2}\).

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