To simplify $(16a^4)^{\frac{3}{4}}$, let's go through the steps:

1. Start by applying the power rule: $(x^m)^n = x^{m \cdot n}$.

   $(16a^4)^{\frac{3}{4}} = 16^{\frac{3}{4}} \cdot (a^4)^{\frac{3}{4}}$

2. Simplify each part separately:

   • For $16^{\frac{3}{4}}$: $16^{\frac{3}{4}} = (2^4)^{\frac{3}{4}} = 2^{4 \cdot \frac{3}{4}} = 2^3 = 8$

   • For $(a^4)^{\frac{3}{4}}$: $(a^4)^{\frac{3}{4}} = a^{4 \cdot \frac{3}{4}} = a^3$

3. Combine the results: $(16a^4)^{\frac{3}{4}} = 8a^3$

The fully simplified expression is:

$\boxed{8a^3}$

Would you like more details, or do you have any questions about this solution?

Here are five related questions to deepen your understanding:

1. How would you simplify $(9b^6)^{\frac{1}{2}}$?
2. What is $(27c^3)^{\frac{2}{3}}$ in simplified form?
3. Can you apply the same steps to simplify $(64x^8)^{\frac{1}{2}}$?
4. How does the power of a fraction $\left(\frac{a}{b}\right)^n$ affect simplification?
5. How would you simplify $(25y^{10})^{\frac{1}{5}}$?

Tip: Remember that fractional exponents represent roots; for example, $x^{\frac{1}{2}}$ is the square root of $x$.