Expanding and Simplifying (ab)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: (ab)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
StepbyStep Simplification

Expanding (ab)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

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

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

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

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:
**(ab)³(a+b)³(a²+b²)³ = ** a¹²  6a⁸b⁴ + 15a⁴b⁸  10b¹²
Conclusion
We successfully expanded and simplified the complex expression (ab)³(a+b)³(a²+b²)³ using key algebraic identities and careful stepbystep multiplication and simplification. The final result is a¹²  6a⁸b⁴ + 15a⁴b⁸  10b¹².