Simplifying Expressions with Exponents: (6x^5y^3)^2
This article will explore how to simplify the expression (6x^5y^3)^2. We will delve into the fundamental rules of exponents and demonstrate the steps involved in solving this problem.
Understanding the Rules of Exponents
Before simplifying the expression, let's review the key rules of exponents that will be used:
 Power of a Product Rule: (ab)^n = a^n * b^n
 Power of a Power Rule: (a^m)^n = a^(m*n)
Simplifying the Expression

Applying the Power of a Product Rule:
 We first apply the Power of a Product Rule to distribute the exponent (2) to each factor within the parentheses:
 (6x^5y^3)^2 = 6^2 * (x^5)^2 * (y^3)^2
 We first apply the Power of a Product Rule to distribute the exponent (2) to each factor within the parentheses:

Applying the Power of a Power Rule:
 Next, we apply the Power of a Power Rule to simplify the exponents of the variables:
 6^2 * (x^5)^2 * (y^3)^2 = 6^2 * x^(52) * y^(32)
 Next, we apply the Power of a Power Rule to simplify the exponents of the variables:

Simplifying:
 Finally, we perform the multiplications to obtain the simplified expression:
 6^2 * x^(52) * y^(32) = 36x^10y^6
 Finally, we perform the multiplications to obtain the simplified expression:
Conclusion
Therefore, the simplified form of (6x^5y^3)^2 is 36x^10y^6. By applying the fundamental rules of exponents, we can effectively simplify complex expressions. Understanding these rules is crucial for working with polynomials and other mathematical concepts.