Let's go through each of these simplifications step by step, using the given tip that states:

$\left(x^{\frac{m}{n}}\right)^{\frac{p}{q}} = x^{\frac{m \cdot p}{n \cdot q}}$

Problem Solutions:

1. Simplify $\left( x^{\frac{3}{4}} \right)^{\frac{2}{5}}$

   Using the formula: $x^{\frac{3 \cdot 2}{4 \cdot 5}} = x^{\frac{6}{20}} = x^{\frac{3}{10}}$

2. Simplify $\left( y^{\frac{7}{3}} \right)^{\frac{1}{4}}$

   Using the formula: $y^{\frac{7 \cdot 1}{3 \cdot 4}} = y^{\frac{7}{12}}$

3. Simplify $\left( z^{\frac{5}{6}} \right)^{\frac{3}{2}}$

   Using the formula: $z^{\frac{5 \cdot 3}{6 \cdot 2}} = z^{\frac{15}{12}} = z^{\frac{5}{4}}$

4. Simplify $\left( a^{\frac{4}{7}} \right)^{\frac{5}{6}}$

   Using the formula: $a^{\frac{4 \cdot 5}{7 \cdot 6}} = a^{\frac{20}{42}} = a^{\frac{10}{21}}$

5. Simplify $\left( b^{\frac{2}{3}} \right)^{\frac{4}{5}}$

   Using the formula: $b^{\frac{2 \cdot 4}{3 \cdot 5}} = b^{\frac{8}{15}}$

Final Answers:

1. $x^{\frac{3}{10}}$
2. $y^{\frac{7}{12}}$
3. $z^{\frac{5}{4}}$
4. $a^{\frac{10}{21}}$
5. $b^{\frac{8}{15}}$

Would you like further details on any specific step, or do you have additional questions?

Here are 5 related questions to explore:

1. How would you simplify $\left(m^{\frac{5}{9}}\right)^{\frac{3}{4}}$?
2. What is the simplified form of $\left(n^{\frac{7}{8}}\right)^{\frac{2}{3}}$?
3. If you have $\left(p^{\frac{6}{5}}\right)^{\frac{5}{6}}$, what is the result?
4. Can you generalize the process for any fractional exponents raised to another power?
5. Why is it beneficial to reduce fractions in exponent expressions?

Tip: Always reduce the fraction in the exponent for a cleaner and simplified final answer.