Simplify each part: (2x²y³)³ = 8x⁶y⁹, (3xy²)² = 9x²y⁴ - Treasure Valley Movers

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.

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(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

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(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

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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.

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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.