Sure! Let's work through the process of rationalizing and simplifying the given expression step-by-step.
We start with the expression:
[tex]\[
\frac{\sqrt{5}}{a - \sqrt{5}}
\][/tex]
Our goal is to rationalize the denominator. This means we want to eliminate the square root from the denominator. To do this, we'll multiply the numerator and denominator by the conjugate of the denominator. The conjugate of [tex]\(a - \sqrt{5}\)[/tex] is [tex]\(a + \sqrt{5}\)[/tex].
So, we multiply by:
[tex]\[
\frac{\sqrt{5}}{a - \sqrt{5}} \cdot \frac{a + \sqrt{5}}{a + \sqrt{5}}
\][/tex]
In the numerator:
[tex]\[
\sqrt{5} \cdot (a + \sqrt{5}) = \sqrt{5} a + 5
\][/tex]
In the denominator, we use the difference of squares formula:
[tex]\[
(a - \sqrt{5})(a + \sqrt{5}) = a^2 - (\sqrt{5})^2 = a^2 - 5
\][/tex]
Putting these together, we have:
[tex]\[
\frac{\sqrt{5} a + 5}{a^2 - 5}
\][/tex]
The simplified form of the given expression is:
[tex]\[
\frac{\sqrt{5} a + 5}{a^2 - 5}
\][/tex]
Thus, by rationalizing the denominator and simplifying, we obtain:
[tex]\[
\frac{\sqrt{5} a + 5}{a^2 - 5}
\][/tex]
