To solve $(5^4)^3$, you apply the power of a power rule, which states that $(a^m)^n = a^{m \cdot n}$. Here, the base is $5$, and the exponents are $4$ and $3$.

$(5^4)^3 = 5^{4 \cdot 3} = 5^{12}$

Now, calculating $5^{12}$:

$5^{12} = 244140625$

So, $(5^4)^3 = 244140625$.

Would you like further details or any clarifications? Here are 5 questions to deepen understanding:

1. How does the power of a power rule work for negative exponents?
2. Can you explain how to solve $(3^2)^4$ using the same method?
3. What would change in the calculation if the base were a fraction?
4. How do you calculate powers of numbers without a calculator?
5. What is the general rule for simplifying $(x^a)^b$?

Tip: The power of a power rule is a shortcut to simplify expressions quickly, avoiding unnecessary intermediate calculations!