To simplify the expression $(x^3 y^4)^2$, follow these steps:

1. Apply the power rule for exponents: $(a^m b^n)^p = a^{m \cdot p} b^{n \cdot p}$.
2. Distribute the exponent $2$ to both $x^3$ and $y^4$.

$(x^3 y^4)^2 = (x^3)^2 (y^4)^2$

Now apply the power of a power rule, which states that $(a^m)^p = a^{m \cdot p}$:

$(x^3)^2 = x^{3 \cdot 2} = x^6$ $(y^4)^2 = y^{4 \cdot 2} = y^8$

Thus, the simplified expression is:

$x^6 y^8$

Would you like more details or have any further questions?

Here are five related questions:

1. How do you simplify $(a^5 b^2)^3$?
2. What is the general rule for powers of a product?
3. How would you simplify $(x^2 y^3 z)^4$?
4. Can you explain the difference between $(x^m)^n$ and $x^{m+n}$?
5. How do you handle negative exponents in similar expressions?

Tip: When raising an expression to a power, always apply the exponent to each base inside the parentheses.