Simplifying Exponential Expressions: (m^5n/pq^2)^4
This article will delve into simplifying the expression (m^5n/pq^2)^4. We will use the rules of exponents to break down the expression and arrive at a simplified form.
Understanding the Rules of Exponents
Before we begin, let's review the key rules of exponents that we'll be applying:
 Power of a Product: (ab)^n = a^n * b^n
 Power of a Quotient: (a/b)^n = a^n / b^n
 Power of a Power: (a^m)^n = a^(m*n)
Simplifying the Expression

Apply the Power of a Quotient rule:
(m^5n/pq^2)^4 = (m^5n)^4 / (pq^2)^4

Apply the Power of a Product rule to both numerator and denominator:
(m^5n)^4 / (pq^2)^4 = (m^5)^4 * (n)^4 / (p)^4 * (q^2)^4

Apply the Power of a Power rule to each term:
(m^5)^4 * (n)^4 / (p)^4 * (q^2)^4 = m^(54) * n^4 / p^4 * q^(24)

Simplify the exponents:
m^(54) * n^4 / p^4 * q^(24) = m^20 * n^4 / p^4 * q^8
Final Result
Therefore, the simplified form of the expression (m^5n/pq^2)^4 is m^20 * n^4 / p^4 * q^8.