Simplifying Expressions with Exponents: (5xy^2)^4
In mathematics, understanding how to simplify expressions with exponents is crucial. One common example involves expressions like (5xy^2)^4. Let's break down the steps to simplify this expression:
Understanding the Rules
 Power of a product: When a product is raised to a power, each factor is raised to that power. This means: (ab)^n = a^n * b^n.
 Power of a power: When a power is raised to another power, the exponents are multiplied. This means: (a^m)^n = a^(m*n).
Applying the Rules to (5xy^2)^4

Apply the power of a product rule:
 (5xy^2)^4 = 5^4 * x^4 * (y^2)^4

Apply the power of a power rule:
 5^4 * x^4 * (y^2)^4 = 5^4 * x^4 * y^(2*4)

Simplify the exponents:
 5^4 * x^4 * y^(2*4) = 625 * x^4 * y^8
Conclusion
Therefore, the simplified form of (5xy^2)^4 is 625x^4y^8.
This process demonstrates how applying the fundamental rules of exponents allows us to simplify complex expressions, making them easier to understand and work with.