%5E3-(a-b)%5E3-formula.jpeg "(a+b)^3-(a-b)^3 Formula")
Understanding the (a+b)^3-(a-b)^3 Formula
The formula (a+b)^3-(a-b)^3 = 6ab(a+b) is a useful algebraic identity that simplifies the expansion of the cubic expressions. It's particularly helpful for solving problems in algebra and calculus.
Derivation of the Formula
We can derive this formula by expanding the cubes using the binomial theorem:
- (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
- (a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3
Now, subtracting the second equation from the first, we get:
(a+b)^3 - (a-b)^3 = (a^3 + 3a^2b + 3ab^2 + b^3) - (a^3 - 3a^2b + 3ab^2 - b^3)
Simplifying the equation, we eliminate the terms that cancel out:
(a+b)^3 - (a-b)^3 = 6a^2b + 6b^3
Finally, factoring out 6ab, we arrive at the desired formula:
(a+b)^3 - (a-b)^3 = 6ab(a+b)
Applications of the Formula
This formula has several applications, including:
- Simplifying complex algebraic expressions: By recognizing this pattern in an expression, you can simplify it and make it easier to work with.
- Solving equations: The formula can be used to solve equations involving cubic expressions.
- Calculating values: It can be used to efficiently calculate the difference of cubes without expanding the entire expression.
Example
Let's say we need to find the value of (5+3)^3 - (5-3)^3. Instead of directly expanding the cubes, we can apply the formula:
(5+3)^3 - (5-3)^3 = 6 * 5 * 3 (5 + 3)
Simplifying further:
= 90 * 8
= 720
Therefore, (5+3)^3 - (5-3)^3 = 720.
Conclusion
The formula (a+b)^3-(a-b)^3 = 6ab(a+b) is a valuable tool in algebra, simplifying calculations and solving equations. Understanding its derivation and applications can be beneficial for students and anyone working with algebraic expressions.