CAT 2023 Slot 3 QA Q4 Logarithms & Exponents

Question 4 of the Quantitative Ability (QA) section, CAT 2023 Slot 3, with CAT's official answer and a worked solution.

If \(x\) is a positive real number such that \(x^8+\left(\frac{1}{x}\right)^8=47\), then the value of \(x^9+\left(\frac{1}{x}\right)^9\) is

A
\(40 \sqrt{5}\)
B
\(36 \sqrt{5}\)
C
\(30 \sqrt{5}\)
D
\(34 \sqrt{5}\)

Answer: (D) 34 √(5) Logarithms & Exponents MCQ

Solution

\(x^8+x^{-8}=47 \Rightarrow (x^4+x^{-4})^2 = 49 \Rightarrow x^4+x^{-4}=7\), then \(x^2+x^{-2}=3\), then \(x+x^{-1}=\sqrt5\) (x > 0).
\(x^3+x^{-3} = s^3-3s\) with \(s=\sqrt5\): \(5\sqrt5-3\sqrt5 = 2\sqrt5\).
\(x^9+x^{-9} = u^3-3u\) with \(u=2\sqrt5\): \(40\sqrt5 - 6\sqrt5 = 34\sqrt5\).

Shortcut. Each squaring step back: 47 to 7 to 3 to \(\sqrt5\); then cube twice using \(a^3+b^3\) with \(ab=1\).