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Power of x expression

CAT 2020 Slot 1 · Quantitative Ability · Hard · Number Theory

This is a hard Quantitative Ability question from the CAT 2020 Slot 1 paper. It tests Number Theory. The full answer key and a step-by-step explanation are below — try it yourself first, then reveal the solution.

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If x=(4096)7+43x = (4096)^{7+4\sqrt{3}}, then which of the following equals 64?

  1. A.

    \frac{x^{7/2}}{x^{4/\sqrt{3}}}

  2. B.

    \frac{x^7}{x^{4\sqrt{3}}}

  3. C.

    \frac{x^{7/2}}{x^{2\sqrt{3}}}

  4. D.

    \frac{x^7}{x^{2\sqrt{3}}}

Answer

C

Explanation

Notice that 4096=6424096 = 64^2. Thus, x=(642)7+43=642(7+43)x = (64^2)^{7+4\sqrt{3}} = 64^{2(7+4\sqrt{3})}.

Taking power 12(7+43)\frac{1}{2(7+4\sqrt{3})} on both sides: 64=x12(7+43)64 = x^{\frac{1}{2(7+4\sqrt{3})}}

Rationalizing the exponent: 12(7+43)=7−432(49−48)=7−432=72−23\frac{1}{2(7+4\sqrt{3})} = \frac{7 - 4\sqrt{3}}{2(49 - 48)} = \frac{7 - 4\sqrt{3}}{2} = \frac{7}{2} - 2\sqrt{3}

So, 64=x7/2−23=x7/2x2364 = x^{7/2 - 2\sqrt{3}} = \frac{x^{7/2}}{x^{2\sqrt{3}}}

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