We need to simplify the expression $(216)^{\frac{2}{3}} \times (0.16)^{-\frac{3}{2}}$.

1. Problem Description

We need to simplify the expression $(216)^{\frac{2}{3}} \times (0.16)^{-\frac{3}{2}}$.

2. Solution Steps

First, we simplify $(216)^{\frac{2}{3}}$. Since $216 = 6^3$, we have:

$(216)^{\frac{2}{3}} = (6^3)^{\frac{2}{3}} = 6^{3 \times \frac{2}{3}} = 6^2 = 36$.

Next, we simplify $(0.16)^{-\frac{3}{2}}$. We can write $0.16$ as $\frac{16}{100} = \frac{4}{25} = (\frac{2}{5})^2$. Then,

$(0.16)^{-\frac{3}{2}} = \left(\frac{4}{25}\right)^{-\frac{3}{2}} = \left(\left(\frac{2}{5}\right)^2\right)^{-\frac{3}{2}} = \left(\frac{2}{5}\right)^{2 \times (-\frac{3}{2})} = \left(\frac{2}{5}\right)^{-3} = \left(\frac{5}{2}\right)^3 = \frac{5^3}{2^3} = \frac{125}{8}$.

Finally, we multiply the simplified terms:

$36 \times \frac{125}{8} = \frac{36 \times 125}{8} = \frac{4500}{8} = \frac{1125}{2} = 562.5$

3. Final Answer

$562.5$