The expression in the image involves cube roots and fractions. Let's break it down step by step:

[ \frac{1}{\sqrt[3]{\frac{8}{0.000216}}} + \sqrt[3]{0.000343} ]

Step 1: Simplify the first term

The first term involves a cube root inside a fraction: $\frac{1}{\sqrt[3]{\frac{8}{0.000216}}}$ We can first simplify the fraction inside the cube root: $\frac{8}{0.000216} \approx 37037.037$ Now, take the cube root: $\sqrt[3]{37037.037} \approx 33.206$ Thus, the first term becomes: $\frac{1}{33.206} \approx 0.0301$

Step 2: Simplify the second term

The second term is the cube root of 0.000343: $\sqrt[3]{0.000343} = 0.07$

Step 3: Combine the results

Now, we add the two simplified terms: $0.0301 + 0.07 = 0.1001$

Final Result:

$\boxed{0.1001}$

Would you like more details on how the calculations were performed? Or do you have any further questions?

Related Questions:

1. How do you calculate cube roots manually?
2. What is the significance of cube roots in real-world applications?
3. How do you simplify cube roots involving fractions?
4. Can cube roots be expressed as exponents? How?
5. What are some practical examples where cube roots are used in geometry?

Tip: When calculating cube roots, it's often helpful to approximate the numbers first and refine the result by using a calculator or logarithms if needed.