Question

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

• A. $\frac{x^7}{x^{2\sqrt3}}$
• B. $\frac{x^7}{x^{4\sqrt3}}$
• C. $\frac{x\frac{7}{2}}{x^{\frac{4}{\sqrt3}}}$
• D. $\frac{x\frac{7}{2}}{x^{2\sqrt3}}$

Answer 1

Answer: (D)

$\frac{x\frac{7}{2}}{x^{2\sqrt3}}$

Answer 2

The correct answer is (D):$\frac{x\frac{7}{2}}{x^{2\sqrt3}}$

$x = (4096)^{7+4√3}$

$\frac{1}{x^{7+4\sqrt3}} = (4096)$

On rationalizing $7+4\sqrt3$, we get $\frac{1}{7+4\sqrt3} = 7-4\sqrt3$

$∴ x^{7-4\sqrt3} = (64)^2$

$∴ 64 = x^{\frac{7-4√3}{2}}$

$64 = \frac{x^{\frac{7}{2}}}{x^{2√3}}$