Question

Rationalize the denominator.

5/ 3+ squareroot 23

5/ 3+ squareroot 23 = blank (Please simplify your answer. Type an exact answer, using radicals as needed.)

Answer 1

To rationalize the denominator of the expression \(\frac{5}{3 + \sqrt{23}}\), we can multiply the numerator and the denominator by the conjugate of the denominator, which is \(3 - \sqrt{23}\).

Here are the steps:

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

\[
\frac{5}{3 + \sqrt{23}} \cdot \frac{3 - \sqrt{23}}{3 - \sqrt{23}} = \frac{5(3 - \sqrt{23})}{(3 + \sqrt{23})(3 - \sqrt{23})}
\]

2. Simplify the denominator using the difference of squares formula:

\[
(3 + \sqrt{23})(3 - \sqrt{23}) = 3^2 - (\sqrt{23})^2 = 9 - 23 = -14
\]

3. Now the expression becomes:

\[
\frac{5(3 - \sqrt{23})}{-14}
\]

4. Distributing the numerator:

\[
\frac{15 - 5\sqrt{23}}{-14} = -\frac{15 - 5\sqrt{23}}{14}
\]

5. This can be split into two parts:

\[
-\frac{15}{14} + \frac{5\sqrt{23}}{14}
\]

Thus, the rationalized expression is:

\[
\frac{5\sqrt{23}}{14} - \frac{15}{14}
\]

In summary, the simplified answer is:

\[
\frac{5\sqrt{23}}{14} - \frac{15}{14}
\]