Let \( x = (4096)^{7 + 4\sqrt{3}} \).
Given that \( 4096 = 2^{12} \), we can write:
\[ x = (2^{12})^{7 + 4\sqrt{3}} = 2^{12(7 + 4\sqrt{3})} = 2^{84 + 48\sqrt{3}} \]
We will now evaluate each option to determine which expression equals \( 64 = 2^6 \).
Option 1: \( \frac{x^7}{x^{2\sqrt{3}}} = x^{7 - 2\sqrt{3}} \)
\[ x^{7 - 2\sqrt{3}} = \left(2^{84 + 48\sqrt{3}}\right)^{7 - 2\sqrt{3}} = 2^{(84 + 48\sqrt{3})(7 - 2\sqrt{3})} \]
Expanding the exponent: \[ 84 \cdot 7 + 84 \cdot (-2\sqrt{3}) + 48\sqrt{3} \cdot 7 + 48\sqrt{3} \cdot (-2\sqrt{3}) \] \[ = 588 - 168\sqrt{3} + 336\sqrt{3} - 288 = 300 + 168\sqrt{3} \]
Therefore, the result is: \[ 2^{300 + 168\sqrt{3}} eq 64 \]
Option 2: \( \frac{x^7}{x^{4\sqrt{3}}} = x^{7 - 4\sqrt{3}} \)
\[ x^{7 - 4\sqrt{3}} = \left(2^{84 + 48\sqrt{3}}\right)^{7 - 4\sqrt{3}} = 2^{(84 + 48\sqrt{3})(7 - 4\sqrt{3})} \]
Expanding the exponent: \[ 84 \cdot 7 + 84 \cdot (-4\sqrt{3}) + 48\sqrt{3} \cdot 7 + 48\sqrt{3} \cdot (-4\sqrt{3}) \] \[ = 588 - 336\sqrt{3} + 336\sqrt{3} - 576 = 12 \]
Thus, the result is: \[ 2^{12} = 4096 eq 64 \]
Option 3: \( \frac{x^{7/2}}{x^{4/\sqrt{3}}} = x^{\frac{7}{2} - \frac{4}{\sqrt{3}}} \)
\[ x^{\frac{7}{2} - \frac{4}{\sqrt{3}}} = \left(2^{84 + 48\sqrt{3}}\right)^{\frac{7}{2} - \frac{4}{\sqrt{3}}} \]
This expression does not simplify to a neat integer power of 2. Therefore, it is also incorrect.
Option 4: \( \frac{x^{7/2}}{x^{2\sqrt{3}}} = x^{\frac{7}{2} - 2\sqrt{3}} \)
\[ x^{\frac{7}{2} - 2\sqrt{3}} = \left(2^{84 + 48\sqrt{3}}\right)^{\frac{7}{2} - 2\sqrt{3}} = 2^{(84 + 48\sqrt{3})(\frac{7}{2} - 2\sqrt{3})} \]
Expanding the exponent: \[ 84 \cdot \frac{7}{2} + 84 \cdot (-2\sqrt{3}) + 48\sqrt{3} \cdot \frac{7}{2} + 48\sqrt{3} \cdot (-2\sqrt{3}) \] \[ = 294 - 168\sqrt{3} + 168\sqrt{3} - 288 = 6 \]
Consequently, the result is: \[ 2^6 = 64 \]
Final Answer: \( \boxed{\frac{x^{7/2}}{x^{2\sqrt{3}}}} \)