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:
- Multiply the numerator and the denominator by the conjugate:
\[ \frac{5}{3 + \sqrt{23}} \cdot \frac{3 - \sqrt{23}}{3 - \sqrt{23}} = \frac{5(3 - \sqrt{23})}{(3 + \sqrt{23})(3 - \sqrt{23})} \]
- Simplify the denominator using the difference of squares formula:
\[ (3 + \sqrt{23})(3 - \sqrt{23}) = 3^2 - (\sqrt{23})^2 = 9 - 23 = -14 \]
- Now the expression becomes:
\[ \frac{5(3 - \sqrt{23})}{-14} \]
- Distributing the numerator:
\[ \frac{15 - 5\sqrt{23}}{-14} = -\frac{15 - 5\sqrt{23}}{14} \]
- 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:
\[ \frac{5\sqrt{23}}{14} - \frac{15}{14} \]
1 answer