Power of x expression

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

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

Rationalizing the exponent: $\frac{1}{2(7+4\sqrt{3})} = \frac{7 - 4\sqrt{3}}{2(49 - 48)} = \frac{7 - 4\sqrt{3}}{2} = \frac{7}{2} - 2\sqrt{3}$

So, $64 = x^{7/2 - 2\sqrt{3}} = \frac{x^{7/2}}{x^{2\sqrt{3}}}$