An algebra problem by Jason Chrysoprase

$\begin{aligned} \sqrt{\frac{5a^2c-4c^2e+e^3}{5b^2d-4d^2f+f^3}} & = \sqrt{\frac{5 \times 64^2 \times b^2 \times 64 \times d-4 \times 64^2 \times d^2 \times 64 \times f + 64^3 \times f^3}{5b^2d-4d^2f+f^3}} \\ & = \sqrt{\frac{64^3(5b^2d-4d^2f+f^3)}{5b^2d-4d^2f+f^3}} \\ & = \sqrt{64^3} \\ & = \sqrt{(2^6)^3} \\ & = \sqrt{2^{18}} \\ &= \sqrt{2^9} \\ &= 512 \\ \end{aligned}$