Simplifying Expressions with Exponents: (4xy^4)^3
In mathematics, understanding how to simplify expressions with exponents is crucial. This article will guide you through the process of simplifying the expression (4xy^4)^3.
Understanding the Rules of Exponents
Before we dive into the simplification, let's review the key exponent rules we'll be using:
 Product of powers: (x^m)(x^n) = x^(m+n)
 Power of a product: (xy)^n = x^n * y^n
 Power of a power: (x^m)^n = x^(m*n)
Simplifying (4xy^4)^3
Now, let's apply these rules to our expression:

Apply the power of a product rule: (4xy^4)^3 = 4^3 * x^3 * (y^4)^3

Apply the power of a power rule: 4^3 * x^3 * (y^4)^3 = 4^3 * x^3 * y^(4*3)

Simplify the exponents: 4^3 * x^3 * y^(4*3) = 64x^3y^12
Therefore, the simplified form of (4xy^4)^3 is 64x^3y^12.
Key Takeaways
 Understanding the rules of exponents is essential for simplifying complex expressions.
 The power of a product rule and the power of a power rule are particularly useful when dealing with expressions raised to a power.
 Simplifying expressions can often be done by breaking them down into smaller parts and applying the appropriate rules.