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CAT 2020 Slot 1 — QA Question 17

IndicesMedium

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

Answer & solution

  • A

    x72x43

  • B

    x7x23

  • x72x23 

  • D

    x7x43

Solution

Medium

Write 40964096 as a power of 6464 to expose the exponent of 6464 inside xx. Then 64=x1/(that exponent)64 = x^{1/(\text{that exponent})}. Rationalise the reciprocal exponent (it contains 3\sqrt3) and split it into a power-of-xx expression to match an option.

1

Express xx as a power of 6464. Since 4096=6424096 = 64^2.

x=(4096)7+43=(642)7+43⇒ x=64 14+83⇒ 64=x114+83\begin{aligned} &x = (4096)^{7+4\sqrt3} = (64^2)^{7+4\sqrt3}\\ &\Rightarrow\ x = 64^{\,14+8\sqrt3}\\ &\Rightarrow\ 64 = x^{\frac{1}{14+8\sqrt3}} \end{aligned}
2

Rationalise the exponent. Multiply numerator and denominator by the conjugate 14−8314-8\sqrt3.

114+83=14−83(14)2−(83)2=14−83196−192=14−834=7−432\begin{aligned} &\frac{1}{14+8\sqrt3}=\frac{14-8\sqrt3}{(14)^2-(8\sqrt3)^2}\\ &=\frac{14-8\sqrt3}{196-192}\\ &=\frac{14-8\sqrt3}{4}=\frac{7-4\sqrt3}{2} \end{aligned}
3

Split into a ratio of powers. Substitute the rationalised exponent from step 2.

64=x7−432=x72⋅x−432=x72⋅x−23=x7/2x23\begin{aligned} &64 = x^{\frac{7-4\sqrt3}{2}} = x^{\frac{7}{2}}\cdot x^{-\frac{4\sqrt3}{2}}\\ &= x^{\frac{7}{2}}\cdot x^{-2\sqrt3}\\ &= \frac{x^{7/2}}{x^{2\sqrt3}} \end{aligned}
64=x7/2x23— option (c)64 = \dfrac{x^{7/2}}{x^{2\sqrt3}}\quad\text{— option (c)}
CAT 2020 Slot 1 QA Q17: If x = (4096) 7+4√3 , then which of the following equals 64? — Solution | TheCATExam