Rationalize the denominator.

Question

StartFraction 5 Over StartRoot 5 EndRoot plus 2 EndFraction
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Part 1
StartFraction 5 Over StartRoot 5 EndRoot plus 2 EndFraction
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​(Type an exact​ answer, using radicals as needed. Simplify your answer. Rationalize all​ denominators.)

Answer 1

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:

1. Multiply the numerator and denominator by the conjugate:

\[ \frac{5}{\sqrt{5} + 2} \cdot \frac{\sqrt{5} - 2}{\sqrt{5} - 2} = \frac{5(\sqrt{5} - 2)}{(\sqrt{5} + 2)(\sqrt{5} - 2)}. \]

2. Calculate the denominator using the difference of squares:

\[ (\sqrt{5} + 2)(\sqrt{5} - 2) = (\sqrt{5})^2 - (2)^2 = 5 - 4 = 1. \]

3. Rewrite the expression:

Now we have:

\[ \frac{5(\sqrt{5} - 2)}{1} = 5(\sqrt{5} - 2). \]

4. 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:

\[ \boxed{5\sqrt{5} - 10}. \]