Simplifying the Expression (4xy^3)(3x^3y^5)
In mathematics, simplifying expressions involves combining like terms and applying the rules of exponents. Let's break down how to simplify the expression (4xy^3)(3x^3y^5).
Understanding the Rules of Exponents
The key to simplifying this expression lies in understanding the following rules of exponents:
 Product of powers: When multiplying powers with the same base, add the exponents. For example, x^m * x^n = x^(m+n).
 Product of coefficients: Multiply the coefficients of the terms as you would any numbers.
Simplifying the Expression

Rearrange the terms: It's helpful to group the coefficients and variables separately for easier multiplication: (4 * 3) * (x * x^3) * (y^3 * y^5)

Apply the product of coefficients: (4 * 3) = 12

Apply the product of powers:
 x * x^3 = x^(1+3) = x^4
 y^3 * y^5 = y^(3+5) = y^8

Combine the simplified terms: 12 * x^4 * y^8 = 12x^4y^8
Conclusion
Therefore, the simplified form of the expression (4xy^3)(3x^3y^5) is 12x^4y^8. This process demonstrates how the rules of exponents allow us to efficiently combine and simplify algebraic expressions.