Let's break down and simplify both expressions.

1. First Expression: $\frac{\frac{4}{3}}{8}$

• This is equivalent to multiplying by the reciprocal: $\frac{4}{3} \div 8 = \frac{4}{3} \times \frac{1}{8} = \frac{4 \times 1}{3 \times 8} = \frac{4}{24} = \frac{1}{6}$

Thus, the simplified result for the first expression is $\frac{1}{6}$.

2. Second Expression: $\left( \frac{1}{125} \right)^{\frac{2}{3}}$

• The cube root of 125 is 5, so we first simplify $\left( \frac{1}{125} \right)^{\frac{1}{3}} = \frac{1}{5}$.
• Now square the result: $\left( \frac{1}{125} \right)^{\frac{2}{3}} = \left( \frac{1}{5} \right)^2 = \frac{1}{25}$

Thus, the simplified result for the second expression is $\frac{1}{25}$.

Final Answers:

1. $\frac{1}{6}$
2. $\frac{1}{25}$

Let me know if you'd like more details or have any questions!

Here are 5 follow-up questions:

1. How do you multiply fractions?
2. What are the rules for dividing by a fraction?
3. How do you calculate the cube root of a number?
4. What are fractional exponents, and how do they work?
5. How can exponentiation with fractions be used to simplify radicals?

Tip: When simplifying complex expressions, try to break them down step by step, simplifying small parts first before recombining them.