Cube Root Squared

$\begin{aligned} X & = \frac 1{\sqrt{\frac {\left(\sqrt[3]{0.000027}\times \sqrt[3]{0.064}\right)^2}{0.000216^{2/3}}}} \\ & = \frac 1{\sqrt{ \left( \frac {0.000027 \times 0.064}{0.000216}\right)^\frac 23}} \\ & = \sqrt[3]{ \frac {0.000216}{0.000027 \times 0.064}} \\ & = \sqrt[3]{ \frac {216}{27 \times 0.064}} \\ & = \sqrt[3]{ \frac {216\times 1000}{27 \times 64}} \\ & = \sqrt[3]{ \frac {6^3\times 10^3}{3^3 \times 4^3}} \\ & = \frac {6\times 10}{3 \times 4} \\ & = \boxed{5} \end{aligned}$

The cube root of 0.000027 is 0.03 and the cube root of 0.064 is 0.4. Multiplying these two numbers yield 0.012 .

$0.012^2$ is equal to 0.000144. 0.000216 to the power of two-thirds can be rewritten as the cube root as 0.000216 to the power of 2, which is 0.0036.

0.000144 over 0.0036 is equal to 0.04. The square root of 0.04 is equal to 0.2. 1 over 0.2 is equal to $\boxed{5}$ .