Simplifying Algebraic Expressions: (a^3b^4c^6)(4ac^3)^2
This article will guide you through simplifying the algebraic expression (a^3b^4c^6)(4ac^3)^2.
Understanding the Rules
Before we dive into the simplification, let's review the key rules we'll use:
 Exponent rule: (x^m)^n = x^(m*n)
 Product rule: (x^m)(x^n) = x^(m+n)
StepbyStep Simplification

Simplify the exponent:
 (4ac^3)^2 = 4^2 * a^2 * (c^3)^2 = 16a^2c^6

Combine the terms:
 (a^3b^4c^6)(16a^2c^6) = 16 * a^3 * a^2 * b^4 * c^6 * c^6

Apply the product rule:
 16a^(3+2) * b^4 * c^(6+6) = 16a^5b^4c^12
Final Result
Therefore, the simplified form of the expression (a^3b^4c^6)(4ac^3)^2 is 16a^5b^4c^12.
Conclusion
Simplifying algebraic expressions involves applying the appropriate rules of exponents and multiplication. By breaking down the expression stepbystep, we can arrive at a simplified and more manageable form. This process is essential in various mathematical fields, particularly in algebra and calculus.