CAT 2020 Slot 1 QA Q9 Logarithms & Exponents

Question 9 of the Quantitative Ability (QA) section, CAT 2020 Slot 1, with CAT's official answer and a worked solution.

If x = (4096)^7+4√3, then which of the following equals 64?

A
\(\frac{x^{7/2}}{x^{4/√3}}\)
B
\(\frac{x^{7}}{x^{4√3}}\)
C
\(\frac{x^{7/2}}{x^{2√3}}\)
D
\(\frac{x^{7}}{x^{2√3}}\)

Answer: (C) x^7/2x^2√3 Logarithms & Exponents MCQ

Solution

\(x = 2^{12(7+4\sqrt3)}\) and \(64 = 2^6\), so we need \(x^k\) with \(12(7+4\sqrt3)k = 6\).
\(k = \frac{1}{2(7+4\sqrt3)} = \frac{7-4\sqrt3}{2}\) (since \((7+4\sqrt3)(7-4\sqrt3)=1\)).
So \(x^{7/2-2\sqrt3} = \frac{x^{7/2}}{x^{2\sqrt3}} = 64\): option C.