%5E4.jpeg "(4x^2y^5)^4")
Simplifying Expressions with Exponents: (4x²y⁵)⁴
This article delves into the simplification of the expression (4x²y⁵)⁴. Understanding the rules of exponents is crucial for efficiently manipulating and solving algebraic expressions.
The Power of a Product Rule
The key to simplifying this expression lies in the power of a product rule:
(ab)ⁿ = aⁿbⁿ
This rule states that when raising a product to a power, you can distribute the power to each factor within the product.
Applying the Rule
Let's apply this rule to our expression:
(4x²y⁵)⁴ = 4⁴ (x²)⁴ (y⁵)⁴
Now, we need to apply another rule:
(aⁿ)ᵐ = aⁿᵐ
This rule states that when raising a power to another power, we multiply the exponents.
Final Simplification
Applying this rule to our expression, we get:
4⁴ (x²)⁴ (y⁵)⁴ = 256x⁸y²⁰
Therefore, the simplified form of (4x²y⁵)⁴ is 256x⁸y²⁰.
Summary
By applying the power of a product rule and the rule for raising a power to another power, we successfully simplified the expression (4x²y⁵)⁴ to 256x⁸y²⁰. This process demonstrates the importance of understanding exponent rules for simplifying complex algebraic expressions.