To rationalize the denominator of the expression
\[ \frac{5}{\sqrt{5} + 2}, \]
we can multiply the numerator and denominator by the conjugate of the denominator, which is \(\sqrt{5} - 2\). This technique eliminates the square root in the denominator.
Let's perform the calculation step-by-step:
- Multiply the numerator and denominator by the conjugate:
\[ \frac{5}{\sqrt{5} + 2} \cdot \frac{\sqrt{5} - 2}{\sqrt{5} - 2} = \frac{5(\sqrt{5} - 2)}{(\sqrt{5} + 2)(\sqrt{5} - 2)}. \]
- Calculate the denominator using the difference of squares:
\[ (\sqrt{5} + 2)(\sqrt{5} - 2) = (\sqrt{5})^2 - (2)^2 = 5 - 4 = 1. \]
- Rewrite the expression:
Now we have:
\[ \frac{5(\sqrt{5} - 2)}{1} = 5(\sqrt{5} - 2). \]
- Distributing the numerator:
Distributing gives:
\[ 5\sqrt{5} - 10. \]
Thus, the rationalized form of the original expression \(\frac{5}{\sqrt{5} + 2}\) is:
\[ 5\sqrt{5} - 10. \]
So the final answer is:
\[ \boxed{5\sqrt{5} - 10}. \]