Rationalize the denominator.

7/2 +squareroot 13

7/2 +squareroot 13 = blank
(Simplify your answer. Please type an exact answer, using radicals as needed.)

1 answer

To rationalize the denominator of the expression \( \frac{7}{2 + \sqrt{13}} \), we multiply the numerator and denominator by the conjugate of the denominator, which is \( 2 - \sqrt{13} \).

So, we have:

\[ \frac{7}{2 + \sqrt{13}} \cdot \frac{2 - \sqrt{13}}{2 - \sqrt{13}} = \frac{7(2 - \sqrt{13})}{(2 + \sqrt{13})(2 - \sqrt{13})} \]

Let's calculate the denominator first:

\[ (2 + \sqrt{13})(2 - \sqrt{13}) = 2^2 - (\sqrt{13})^2 = 4 - 13 = -9 \]

Now, let's calculate the numerator:

\[ 7(2 - \sqrt{13}) = 14 - 7\sqrt{13} \]

Putting it all together, we get:

\[ \frac{14 - 7\sqrt{13}}{-9} = -\frac{14 - 7\sqrt{13}}{9} = -\frac{14}{9} + \frac{7\sqrt{13}}{9} \]

Thus, the rationalized form of the expression is:

\[ -\frac{14}{9} + \frac{7\sqrt{13}}{9} \]

This can also be written as:

\[ \frac{7\sqrt{13}}{9} - \frac{14}{9} \]

So, your final answer is:

\[ \frac{7\sqrt{13}}{9} - \frac{14}{9} \]