Simplify each part: (2x²y³)³ = 8x⁶y⁹, (3xy²)² = 9x²y⁴
Understanding Exponent Rules: Simplify Each Part Step by Step
When working with algebraic expressions involving exponents, simplification relies on core rules of exponents. This article breaks down two key expressions—(2x²y³)³ and (3xy²)²—step by step, showing how to simplify each correctly using exponent properties. By mastering these fundamentals, you’ll gain clarity and confidence in handling more complex algebraic operations.
(2x²y³)³ Simplified: 8x⁶y⁹
Simplifying (2x²y³)³ involves applying the power of a product rule for exponents. This rule states: (ab)ⁿ = aⁿbⁿ which means raise each factor in the product to the exponent independently.
Step 1: Apply the exponent to each term (2x²y³)³ = 2³ × (x²)³ × (y³)³
Step 2: Calculate powers
- 2³ = 8
- (x²)³ = x² × x² × x² = x⁶ (add exponents when multiplying like bases)
- (y³)³ = y³ × y³ × y³ = y⁹
Step 3: Combine all parts 2³ × (x²)³ × (y³)³ = 8x⁶y⁹
✅ Final simplified expression: 8x⁶y⁹
(3xy²)² Simplified: 9x²y⁴
To simplify (3xy²)², we again use the power of a product rule, but now for a coefficient and multiple variables.
Step 1: Apply the exponent to every component (3xy²)² = 3² × x² × (y²)²
Step 2: Calculate each term
- 3² = 9
- x² stays as x² (exponent remains unchanged)
- (y²)² = y² × y² = y⁴ (add exponents)
Step 3: Multiply simplified parts 3² × x² × (y²)² = 9x²y⁴
✅ Final simplified expression: 9x²y⁴
Why Simplifying Exponents Matters
Simplifying expressions like (2x²y³)³ and (3xy²)² reinforces the importance of exponent rules such as:
- (ab)ⁿ = aⁿbⁿ (power of a product)
- (a^m)ⁿ = a^(mn) (power of a power)
- Multiplication of like bases: xᵐ × xⁿ = x⁽ᵐ⁺ⁿ⁾
Mastering these rules ensures accuracy in algebra and helps with solving equations, expanding polynomials, and working with more advanced math topics.
Key Takeaways:
- Use (ab)ⁿ = aⁿbⁿ to distribute exponents over products.
- For powers of powers: multiply exponents (e.g., (x²)³ = x⁶).
- Keep coefficients separate and simplify variables using exponent rules.
Simplifying (2x²y³)³ to 8x⁶y⁹ and (3xy²)² to 9x²y⁴ is straightforward when applying exponent rules step by step. Keep practicing to boost your algebraic fluency!
