Simplify \((216)^{-\frac{2}{3}} \times (0.16)^{-\frac{3}{2}}\)

We can simplify this expression using the rules of exponents. First, let's simplify \((216)^{-\frac{2}{3}}\). We know that \((216)^{\frac{2}{3}}\) is the cube root of 216 squared. So, \((216)^{-\frac{2}{3}}\) is simply the reciprocal of that value. \[(216)^{-\frac{2}{3}} = \frac{1}{(216)^{\frac{2}{3}}} = \frac{1}{(6^3)^{\frac{2}{3}}} = \frac{1}{6^2}\] Next, let's simplify \((0.16)^{-\frac{3}{2}}\). We can rewrite \(0.16\) as \(\frac{16}{100}\), and then apply the exponent to both the numerator and denominator: \[(0.16)^{-\frac{3}{2}} = \left(\frac{16}{100}\right)^{-\frac{3}{2}} = \frac{(100)^{\frac{3}{2}}}{(16)^{\frac{3}{2}}}\] Now, we can multiply these two simplified expressions: \[\frac{1}{6^2} \times \frac{(100)^{\frac{3}{2}}}{(16)^{\frac{3}{2}}} = \frac{1}{36} \times \frac{1000}{64} = \frac{125}{288}\] Therefore, the answer is \(\frac{125}{288}\).