Simplifying (5a^3b^4)^2
This article will guide you through the process of simplifying the expression (5a^3b^4)^2.
Understanding the Rules
To simplify this expression, we need to utilize a few key rules of exponents:
 Negative exponents: A term raised to a negative exponent is equivalent to its reciprocal raised to the positive version of that exponent. In other words, x^n = 1/x^n.
 Power of a product: When a product is raised to an exponent, each factor within the product is raised to that exponent. This is expressed as (xy)^n = x^n * y^n.
 Power of a power: When a term raised to an exponent is further raised to another exponent, the exponents are multiplied. This is expressed as (x^m)^n = x^(m*n).
Applying the Rules

Address the negative exponent: Begin by rewriting the expression using the negative exponent rule: (5a^3b^4)^2 = 1 / (5a^3b^4)^2

Apply the power of a product rule: Distribute the exponent of 2 to each factor within the parentheses: 1 / (5a^3b^4)^2 = 1 / (5^2 * (a^3)^2 * (b^4)^2)

Apply the power of a power rule: Multiply the exponents within the parentheses: 1 / (5^2 * (a^3)^2 * (b^4)^2) = 1 / (5^2 * a^(32) * b^(42))

Simplify: 1 / (5^2 * a^(32) * b^(42)) = 1 / (25a^6b^8)
Conclusion
By applying the rules of exponents, we have successfully simplified the expression (5a^3b^4)^2 to 1 / (25a^6b^8). Remember to always carefully follow the order of operations and utilize the appropriate exponent rules when simplifying such expressions.