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

The Correct Option is D

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 \(\frac{1}{7+4\sqrt3} = 7-4\sqrt3\)

\(∴ x^{7-4\sqrt3} = (64)^2\)

\(∴ 64 = x^{\frac{7-4√3}{2}}\)

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

Answer 2

Given,
\(x = (4096)^{7+4√3}\)
Simplify,
\(x = (2)^{12(7+4√3)}\)
For \(x^{\frac{7}{2}}\)

\(x^{\frac{7}{2}} = (2)^{\frac{7}{2}\cdot12(7+4√3)}\)

\(x^{\frac{7}{2}} = (2)^{42(7+4√3)}\)
Now, for \(x^{2\sqrt3}\)
\(x^{2\sqrt3} = (2)^{2\sqrt3\cdot12(7+4√3)}\)
\(x^{2\sqrt3} = (2)^{24\sqrt3(7+4√3)}\)

Now put both of them together:
\(\frac{x^{\frac{7}{2}}}{x^{2\sqrt3}} = \frac{2^{42(7+4√3)}}{2^{24\sqrt3(7+4√3)}}\)

\(\frac{x^{\frac{7}{2}}}{x^{2\sqrt3}} = 2^{42(7+4√3)-(24\sqrt3(7+4√3))}\)

\(\frac{x^{\frac{7}{2}}}{x^{2\sqrt3}} = 2^{(7+4√3)(42-24\sqrt3)}\)

\(\frac{x^{\frac{7}{2}}}{x^{2\sqrt3}} = 2^{(7+4√3)(7-4\sqrt3)6}\)

If you solve \((7+4√3)(7-4\sqrt3)\) answer is 1
\(\frac{x^{\frac{7}{2}}}{x^{2\sqrt3}} = 2^6\)

\(\frac{x^{\frac{7}{2}}}{x^{2\sqrt3}} = 64\)

So, the correct option is (D): \(\frac{x^{\frac{7}{2}}}{x^{2\sqrt3}}\)