Simplify. Rationalize the denominator.

Question

5
3+
7

Answer 1

To simplify the expression \( \frac{5}{3 + \sqrt{7}} \) and rationalize the denominator, you can multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 3 + \sqrt{7} \) is \( 3 - \sqrt{7} \).

Here’s how to do it:

1. Multiply the numerator and the denominator by the conjugate:

\[ \frac{5}{3 + \sqrt{7}} \cdot \frac{3 - \sqrt{7}}{3 - \sqrt{7}} = \frac{5(3 - \sqrt{7})}{(3 + \sqrt{7})(3 - \sqrt{7})} \]

2. Simplify the denominator using the difference of squares:

\[ (3 + \sqrt{7})(3 - \sqrt{7}) = 3^2 - (\sqrt{7})^2 = 9 - 7 = 2 \]

So, the denominator becomes 2.

3. Now, distribute in the numerator:

\[ 5(3 - \sqrt{7}) = 15 - 5\sqrt{7} \]

4. Combine everything:

\[ \frac{15 - 5\sqrt{7}}{2} \]

5. This can be separated as:

\[ \frac{15}{2} - \frac{5\sqrt{7}}{2} \]

Thus, the final simplified expression is:

\[ \frac{15}{2} - \frac{5\sqrt{7}}{2} \]