Answer
See the explanation
Work Step by Step
Rationalizing a denominator means to perform an arithmetic operation to a fraction with an irrational denominator so that the denominator becomes a rational number. We multiply both numerator and denominator by the conjugate of the denominator.
$\frac{1}{\sqrt 5}$ has an irrational denominator. The conjugate of $\sqrt 5$ is $\sqrt 5$. We rationalize the denominator:
\begin{equation}
\begin{aligned}
\frac{1}{\sqrt{5}} &= \frac{1}{\sqrt{5}}\cdot \frac{\sqrt{5}}{\sqrt{5}}\\
& =\frac{\sqrt{5}}{5}\\
\end{aligned}
\end{equation}
The fraction $\frac{1}{5+\sqrt 5}$ has an irrational denominator. Its conjugate is $5-\sqrt 5$. We have:
\begin{equation}
\begin{aligned}
\frac{1}{5+\sqrt{5}}&= \frac{1}{5+\sqrt{5}}\cdot \frac{5-\sqrt{5}}{5-\sqrt{5}}\\
& =\frac{5-\sqrt{5}}{25-5}\\
& =\frac{5-\sqrt{5}}{20}\\
\end{aligned}
\end{equation}
