Simplifying (5x²y⁴)⁵
This expression involves raising a monomial to a power. Let's break down the simplification process stepbystep:
Understanding the Power of a Product Rule
The core principle we use here is the power of a product rule: (ab)ⁿ = aⁿbⁿ. This rule states that when raising a product to a power, we raise each factor in the product to that power.
Applying the Rule

Apply the power of a product rule: (5x²y⁴)⁵ = 5⁵ (x²)⁵ (y⁴)⁵

Simplify each factor:
 5⁵ = 3125
 (x²)⁵ = x¹⁰ (using the power of a power rule: (aᵐ)ⁿ = aᵐⁿ)
 (y⁴)⁵ = y²⁰

Combine the simplified factors: 3125x¹⁰y²⁰
Final Result
Therefore, the simplified form of (5x²y⁴)⁵ is 3125x¹⁰y²⁰.