To rationalize the denominator [tex]\(2\sqrt{3} - \sqrt{5}\)[/tex] in the expression [tex]\(\frac{1}{2\sqrt{3} - \sqrt{5}}\)[/tex], we must multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(2\sqrt{3} - \sqrt{5}\)[/tex] is [tex]\(2\sqrt{3} + \sqrt{5}\)[/tex].
Here are the steps:
1. Identify the conjugate:
The conjugate of [tex]\(2\sqrt{3} - \sqrt{5}\)[/tex] is [tex]\(2\sqrt{3} + \sqrt{5}\)[/tex].
2. Multiply by the conjugate:
We multiply the numerator and the denominator of [tex]\(\frac{1}{2\sqrt{3} - \sqrt{5}}\)[/tex] by the conjugate [tex]\(2\sqrt{3} + \sqrt{5}\)[/tex]:
[tex]\[
\frac{1}{2\sqrt{3} - \sqrt{5}} \times \frac{2\sqrt{3} + \sqrt{5}}{2\sqrt{3} + \sqrt{5}} = \frac{2\sqrt{3} + \sqrt{5}}{(2\sqrt{3} - \sqrt{5})(2\sqrt{3} + \sqrt{5})}
\][/tex]
3. Simplify the denominator:
The denominator is a difference of squares:
[tex]\[
(2\sqrt{3})^2 - (\sqrt{5})^2 = 4 \cdot 3 - 5 = 12 - 5 = 7
\][/tex]
4. Write the rationalized expression:
Putting it all together, we get:
[tex]\[
\frac{2\sqrt{3} + \sqrt{5}}{7}
\][/tex]
The rationalization factor, which we used to multiply both the numerator and the denominator, is [tex]\(2\sqrt{3} + \sqrt{5}\)[/tex].
Therefore, the rationalization factor for [tex]\(\frac{1}{2\sqrt{3} - \sqrt{5}}\)[/tex] is:
[tex]\[
\sqrt{5} + 2\sqrt{3}
\][/tex]
