The problem consists of three mathematical expressions that involve roots and powers. Let's solve each one step by step.

1. $\sqrt[3]{0.064 \cdot 343}$

We can split the multiplication:

$0.064 = \left(\frac{4}{10}\right)^3 = \left(0.4\right)^3$ $343 = 7^3$

Now we can simplify the expression inside the cube root:

$0.064 \cdot 343 = \left(0.4\right)^3 \cdot 7^3 = (0.4 \cdot 7)^3 = 2.8^3$

Thus, the cube root simplifies to:

$\sqrt[3]{2.8^3} = 2.8$

So, the result of the first expression is $2.8$.

2. $\sqrt[5]{\frac{7^5}{2^{10}}}$

We can simplify the expression inside the fifth root:

[ \sqrt[5]{\frac{7^5}{2^{10}}} = \frac{\sqrt[5]{7^5}}{\sqrt[5]{2^{10}}} ]

The fifth root of $7^5$ simplifies to $7$, and the fifth root of $2^{10}$ simplifies to:

$\sqrt[5]{2^{10}} = 2^{\frac{10}{5}} = 2^2 = 4$

Thus, the expression becomes:

$\frac{7}{4}$

So, the result of the second expression is $\frac{7}{4}$.

3. $\sqrt[8]{\frac{2^{24} \cdot 3^{16}}{5^{16}}}$

We can simplify the expression inside the eighth root:

$\frac{2^{24} \cdot 3^{16}}{5^{16}} = 2^{24} \cdot \left(\frac{3}{5}\right)^{16}$

Now, taking the eighth root of the powers:

$\sqrt[8]{2^{24}} = 2^{\frac{24}{8}} = 2^3 = 8$

$\sqrt[8]{\left(\frac{3}{5}\right)^{16}} = \left(\frac{3}{5}\right)^2 = \frac{9}{25}$

So, the expression becomes:

$8 \cdot \frac{9}{25} = \frac{72}{25}$

Thus, the result of the third expression is $\frac{72}{25}$.

Final Answers:

1. $2.8$
2. $\frac{7}{4}$
3. $\frac{72}{25}$

Let me know if you want more details or have any questions!

Here are five related questions:

1. How do you simplify powers within roots?
2. What is the general method for solving cube roots of products?
3. How do fractional exponents relate to roots?
4. Can you explain how to handle roots of fractions?
5. How do different bases affect the simplification process in roots?

Tip: When simplifying roots of products or quotients, always look for patterns in the exponents to make the calculations easier.