The given expression to simplify is:

$\left( \frac{2t^5u^3}{4t^4} \right)^3$

Step-by-step Simplification:

1. Simplify the fraction inside the parentheses:

   • The coefficient part: $\frac{2}{4} = \frac{1}{2}$
   • The $t$-term: $\frac{t^5}{t^4} = t^{5-4} = t^1 = t$
   • The $u$-term remains $u^3$.

   So, the fraction becomes: $\frac{t u^3}{2}$

2. Raise the simplified fraction to the power of 3: $\left( \frac{t u^3}{2} \right)^3$

   Apply the power rule: $(\frac{a}{b})^n = \frac{a^n}{b^n}$:

   • Numerator: $(t u^3)^3 = t^3 (u^3)^3 = t^3 u^9$
   • Denominator: $2^3 = 8$

   The result is: $\frac{t^3 u^9}{8}$

Final Simplified Form:

$\boxed{\frac{t^3 u^9}{8}}$

Would you like further clarification or details?

Here are some related questions to expand your understanding:

1. How do you simplify expressions with negative exponents?
2. What are the rules for raising fractions to a power?
3. How do you simplify expressions with multiple variables?
4. What happens when a term in the denominator is raised to a power?
5. How would the solution change if the exponent were negative?

Tip: Always simplify the base before applying powers to make calculations easier!