%5E3-(a-b)%5E3.jpeg "(a+b)^3-(a-b)^3")
Simplifying the Expression (a + b)³ - (a - b)³
This article will explore the simplification of the expression (a + b)³ - (a - b)³. We'll break down the process step-by-step and explore the underlying concepts.
Understanding the Problem
The expression involves the cube of two binomial terms, where the only difference is the sign connecting 'a' and 'b'. This pattern suggests a potential simplification using algebraic identities.
Applying Algebraic Identities
We can utilize the following identities to simplify the expression:
- (x + y)³ = x³ + 3x²y + 3xy² + y³
- (x - y)³ = x³ - 3x²y + 3xy² - y³
Let's apply these identities to our expression:
**(a + b)³ - (a - b)³ = (a³ + 3a²b + 3ab² + b³) - (a³ - 3a²b + 3ab² - b³) **
Simplifying the Expression
Now, we can remove the parentheses and combine like terms:
**(a³ + 3a²b + 3ab² + b³) - (a³ - 3a²b + 3ab² - b³) = a³ + 3a²b + 3ab² + b³ - a³ + 3a²b - 3ab² + b³ **
Notice that 'a³' and '-a³' cancel out, as do '3ab²' and '-3ab²'. This leaves us with:
= 6a²b + 2b³
Final Result
The simplified form of the expression (a + b)³ - (a - b)³ is 6a²b + 2b³.
Key Takeaways
This exercise demonstrates the power of algebraic identities in simplifying complex expressions. By recognizing patterns and applying these identities, we can achieve a more concise and manageable form. Remember, understanding these concepts will be beneficial for further mathematical explorations.