Question

If \[ x=5+\sqrt{24} \] then \[ \sqrt{x}-\frac{1}{\sqrt{x}}= \]

• A. \(\sqrt5\)
• B. \(2\sqrt5\)
• C. \(\sqrt2\)
• D. \(2\sqrt2\)

Hint:

Expressions of the form \(a+2\sqrt b\) can often be written as \((\sqrt m+\sqrt n)^2\).

Answer 1

The Correct Option is D

Concept: Express \(x\) as a perfect square form.

Step 1: Simplify \(x\).
Given: \[ x=5+\sqrt{24} \] \[ =5+2\sqrt6 \] Now compare with: \[ (\sqrt a+\sqrt b)^2=a+b+2\sqrt{ab} \] So: \[ a+b=5 \] \[ ab=6 \] Numbers are: \[ 2,3 \] Thus: \[ x=(\sqrt2+\sqrt3)^2 \] So: \[ \sqrt x=\sqrt2+\sqrt3 \]

Step 2: Find reciprocal.
\[ \frac1{\sqrt x}=\frac1{\sqrt2+\sqrt3} \] Rationalizing: \[ =\frac{\sqrt3-\sqrt2}{(\sqrt2+\sqrt3)(\sqrt3-\sqrt2)} \] \[ =\sqrt3-\sqrt2 \]

Step 3: Subtract.
\[ \sqrt x-\frac1{\sqrt x} \] \[ =(\sqrt2+\sqrt3)-(\sqrt3-\sqrt2) \] \[ =2\sqrt2 \] Thus, the required value is: \[ \boxed{2\sqrt2} \]