%5E3(2m%5E2n%5E5p)%2F6m%5E4n%5E9p%5E8.jpeg "(3m^4n)^3(2m^2n^5p)/6m^4n^9p^8")
Simplifying Algebraic Expressions: A Step-by-Step Guide
This article aims to guide you through the simplification of the algebraic expression: (3m⁴n)³(2m²n⁵p) / 6m⁴n⁹p⁸. We will break down the process step-by-step to ensure clarity and understanding.
Understanding the Properties
Before we begin simplifying, let's refresh our memory on some fundamental algebraic properties:
- Power of a Product: (ab)ⁿ = aⁿbⁿ
- Power of a Power: (aⁿ)ᵐ = aⁿᵐ
- Division of Exponents with the Same Base: aⁿ / aᵐ = aⁿ⁻ᵐ
Step-by-Step Simplification
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Simplify the Cube:
- Apply the power of a product property to (3m⁴n)³:
- (3m⁴n)³ = 3³ (m⁴)³ (n)³ = 27m¹²n³
- Apply the power of a product property to (3m⁴n)³:
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Combine the Numerator:
- Multiply the simplified term with the remaining part of the numerator:
- 27m¹²n³ * 2m²n⁵p = 54m¹⁴n⁸p
- Multiply the simplified term with the remaining part of the numerator:
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Simplify the Denominator:
- There are no further simplifications needed for the denominator.
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Divide the Terms:
- Now, divide the simplified numerator by the denominator:
- (54m¹⁴n⁸p) / (6m⁴n⁹p⁸) = 9m¹⁰n⁻¹p⁻⁷
- Now, divide the simplified numerator by the denominator:
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Express with Positive Exponents:
- To express the final answer with positive exponents, use the rule for dividing exponents with the same base:
- 9m¹⁰n⁻¹p⁻⁷ = 9m¹⁰ / np⁷
- To express the final answer with positive exponents, use the rule for dividing exponents with the same base:
Final Answer
The simplified form of the algebraic expression (3m⁴n)³(2m²n⁵p) / 6m⁴n⁹p⁸ is 9m¹⁰ / np⁷.
By following these steps, you can effectively simplify complex algebraic expressions. Remember to utilize the fundamental properties of exponents to guide your calculations.