The expression \((a^5b^6)^4\) can be simplified using the power of a product property, which states \((xy)^n = x^n y^n\). You also apply the rule for exponents, \((x^m)^n = x^{m \cdot n}\).

Let's simplify step by step:

\[
(a^5b^6)^4 = (a^5)^4 \cdot (b^6)^4
\]

Now apply the exponent rule \((x^m)^n = x^{m \cdot n}\) to both terms:

\[
= a^{5 \cdot 4} \cdot b^{6 \cdot 4}
\]

\[
= a^{20} \cdot b^{24}
\]

Thus, the simplified expression is:

\[
a^{20}b^{24}
\]

Would you like more details or have any questions?

Here are 5 related questions to expand on this concept:
1. How would you simplify \((a^3b^2)^5\)?
2. What is the result of \((x^4y^5)^3\)?
3. How do you simplify \((a^2b^3c^4)^2\)?
4. Can you explain the difference between \((ab)^n\) and \(a^n \cdot b^n\)?
5. How would you simplify \((a^7b^3)^6\)?

**Tip**: When simplifying expressions with exponents, always apply the power rule carefully to both the base and its exponent!

Learn how to simplify the exponential expression (a^5b^6)^4 using the power of a product and exponentiation rules. This solution breaks down key algebraic concepts and is suitable for students in Grades 8-10.

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