To simplify the expression \((4b^2c)^3\), apply the power of a product rule, which states \((abc)^n = a^n b^n c^n\).

1. First, apply the cube (power of 3) to each factor inside the parentheses:
\[
(4b^2c)^3 = 4^3 \cdot (b^2)^3 \cdot c^3
\]

2. Now calculate each power separately:
- \(4^3 = 64\)
- \((b^2)^3 = b^{2 \times 3} = b^6\)
- \(c^3 = c^3\)

3. Combine these results:
\[
(4b^2c)^3 = 64b^6c^3
\]

So, the simplified expression is:
\[
64b^6c^3
\]

Would you like more details or further clarification?

Here are some related questions:
1. How do you simplify expressions with negative exponents?
2. What is the power of a quotient rule in exponents?
3. How do you simplify expressions with fractional exponents?
4. How do you factor out common terms in polynomial expressions?
5. What are the properties of exponents for radicals?

**Tip:** Always remember to distribute the exponent to every factor inside the parentheses when simplifying.

Learn how to simplify the algebraic expression (4b^2c)^3 by applying the power of a product rule. This step-by-step guide explores key exponent rules and demonstrates how to calculate powers efficiently. Suitable for students in Grades 8-10.

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