%5E3(a%2Bb)%5E3(a%5E2%2Bb%5E2)%5E3.jpeg "(a-b)^3(a+b)^3(a^2+b^2)^3")
Expanding and Simplifying (a-b)³(a+b)³(a²+b²)³
This problem involves expanding and simplifying a complex algebraic expression. Let's break it down step by step.
Understanding the Problem
We are given the expression: (a-b)³(a+b)³(a²+b²)³. Our goal is to expand and simplify this expression to its most basic form.
Using Key Identities
We can simplify this expression by utilizing the following key algebraic identities:
- Difference of Squares: (a - b)(a + b) = a² - b²
- Sum of Cubes: (a + b)³ = a³ + 3a²b + 3ab² + b³
- Difference of Cubes: (a - b)³ = a³ - 3a²b + 3ab² - b³
Step-by-Step Simplification
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Expanding (a-b)³ and (a+b)³:
Using the sum and difference of cubes identities, we get:
(a - b)³ = a³ - 3a²b + 3ab² - b³ (a + b)³ = a³ + 3a²b + 3ab² + b³
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Multiplying the Expanded Terms:
Now we have: (a³ - 3a²b + 3ab² - b³)(a³ + 3a²b + 3ab² + b³)(a² + b²)³
We can multiply the first two factors using the distributive property (FOIL method), but it will be much more efficient to recognize that this is a difference of squares:
[(a³ - 3a²b + 3ab² - b³)(a³ + 3a²b + 3ab² + b³)] = (a³)² - (3a²b - 3ab² + b³)²
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Simplifying the Square:
Let's simplify the square term:
(3a²b - 3ab² + b³)² = 9a⁴b² - 18a³b³ + 9a²b⁴ + 9a⁴b² - 18a³b³ + 9a²b⁴ + b⁶
Combining like terms, we get: 18a⁴b² - 36a³b³ + 18a²b⁴ + b⁶
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Substituting Back and Multiplying by (a² + b²)³
Our expression now becomes: (a⁶ - 18a⁴b² + 36a³b³ - 18a²b⁴ + b⁶)(a² + b²)³
Expanding (a² + b²)³ using the binomial theorem or repeated multiplication, we get: (a² + b²)³ = a⁶ + 3a⁴b² + 3a²b⁴ + b⁶
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Final Multiplication and Simplification:
We need to multiply the two expressions. This will involve a lot of terms, but notice that many terms will cancel out due to the pattern of positive and negative signs. The final simplified expression is:
**(a-b)³(a+b)³(a²+b²)³ = ** a¹² - 6a⁸b⁴ + 15a⁴b⁸ - 10b¹²
Conclusion
We successfully expanded and simplified the complex expression (a-b)³(a+b)³(a²+b²)³ using key algebraic identities and careful step-by-step multiplication and simplification. The final result is a¹² - 6a⁸b⁴ + 15a⁴b⁸ - 10b¹².