CAT 2024 Slot 2 QA Q13 Logarithms & Exponents

Question 13 of the Quantitative Ability (QA) section, CAT 2024 Slot 2, with CAT's official answer and a worked solution.

If \((x+6 \sqrt{2})^{\frac{1}{2}}-(x-6 \sqrt{2})^{\frac{1}{2}}=2 \sqrt{2}\), then \(x\) equals

TITA: type in the answer. There are no options and no negative marking.

Answer: 11 Logarithms & Exponents TITA

Solution

Let \(u = \sqrt{x+6\sqrt2}\), \(v = \sqrt{x-6\sqrt2}\). Then \(u - v = 2\sqrt2\) and \(u^2 - v^2 = 12\sqrt2\).
So \(u + v = \frac{12\sqrt2}{2\sqrt2} = 6\), giving \(u = 3 + \sqrt2\).
\(x = u^2 - 6\sqrt2 = (11 + 6\sqrt2) - 6\sqrt2 = 11\).

Shortcut. Divide the difference of squares by the given difference to get the sum.