Fraction Simplification

$\dfrac {4^6}{\sqrt{225\times 36}} = \dfrac {2^{12}}{\sqrt{15^2\times 6^2}} = \dfrac {2^{10}\times 2^2}{15\times 6} = \dfrac {1024\times 4}{90} = \boxed{\dfrac {4096}{90}}$

Assuming you're doing it mentally:

This will give you your numerator:

$4 \times 4 = 16 = 4^2$

$16 \times 4 = 64 = 4^3$

$64 \times 4 = (60 \times 4) + (4\times 4) = 256 = 4^4$

$256 \times 4 = (200 \times 4) + (50 \times 4) + (6 \times 4)= 1,024 = 4^5$

$1,024 \times 4 = (1,000 \times 4) + (20 \times 4) + (4 \times 4) = 4,096 = 4^6$

This will give you your denominator:

You can find the square roots individually in $\sqrt{225 \times 36}$ . $\sqrt{225}$ = 15 and $\sqrt{36}$ = 6. From there, you simply multiply $15 \times 6$ , giving you the result of 90.

This gives you the answer of $\frac{4,096}{90}$