Question

Calculate the value of:
\[ \frac{\sqrt{\sqrt{5}+2} + \sqrt{\sqrt{5}-2}}{\sqrt{\sqrt{5}+1}} \]

• A. 1
• B. $\sqrt{2}$
• C. 4
• D. $\sqrt{5}$

Hint:

Whenever you see expressions like \( \sqrt{a+b} + \sqrt{a-b} \), squaring the expression often makes simplification easy and avoids complex algebra.

Answer 1

The Correct Option is B

Step 1: Let us analyze the given expression carefully.
The numerator consists of two square root terms involving \( \sqrt{5} \), and the denominator also involves a square root expression. Our aim is to simplify the expression step by step.
Step 2: Observe a useful identity.
For expressions of the form \( \sqrt{a+b} + \sqrt{a-b} \), squaring the expression helps in simplification.
Step 3: Let
\[ x = \sqrt{\sqrt{5}+2} + \sqrt{\sqrt{5}-2} \]
Step 4: Square the value of \(x\).
\[ x^2 = (\sqrt{\sqrt{5}+2} + \sqrt{\sqrt{5}-2})^2 \]
\[ x^2 = (\sqrt{5}+2) + (\sqrt{5}-2) + 2\sqrt{(\sqrt{5}+2)(\sqrt{5}-2)} \]
\[ x^2 = 2\sqrt{5} + 2\sqrt{5-4} \]
\[ x^2 = 2\sqrt{5} + 2\sqrt{1} \]
\[ x^2 = 2\sqrt{5} + 2 \]
Step 5: Take square root.
\[ x = \sqrt{2(\sqrt{5}+1)} \]
Step 6: Substitute back into the original expression.
\[ \frac{\sqrt{2(\sqrt{5}+1)}}{\sqrt{\sqrt{5}+1}} \]
Step 7: Simplify the fraction.
\[ = \sqrt{2} \]
Step 8: Final conclusion.
Hence, the value of the given expression is \(\sqrt{2}\).