%5E-2-formula.jpeg "(a-b)^-2 Formula")
(a-b)^-2 Formula: A Comprehensive Guide
In mathematics, the formula for raising a binomial expression to a power is a fundamental concept in algebra. One of the most important formulas in this context is the (a-b)^-2 formula, which is widely used in various mathematical operations. In this article, we will explore the (a-b)^-2 formula, its derivation, and some examples to illustrate its application.
Derivation of the (a-b)^-2 Formula
The (a-b)^-2 formula is derived from the binomial theorem, which states that:
(a+b)^n = a^n + na^(n-1)b + (n(n-1))/2! a^(n-2)b^2 + … + b^n
where n is a positive integer.
To derive the (a-b)^-2 formula, we can start with the binomial theorem and substitute -b for b:
(a-(-b))^n = a^n + na^(n-1)(-b) + (n(n-1))/2! a^(n-2)(-b)^2 + … + (-b)^n
Simplifying the expression, we get:
(a-b)^n = a^n - na^(n-1)b + (n(n-1))/2! a^(n-2)b^2 - … + (-1)^n b^n
Now, we can substitute -2 for n to get the (a-b)^-2 formula:
(a-b)^-2 = a^(-2) + (-2)a^(-3)b + ((-2)(-3))/2! a^(-4)b^2 + … + b^(-2)
Simplifying the expression, we get:
(a-b)^-2 = 1/a^2 + 2b/a^3 + 3b^2/a^4 + …
Examples and Applications
Example 1:
Simplify the expression (x-2)^-2.
Using the (a-b)^-2 formula, we get:
(x-2)^-2 = 1/x^2 + 2(2)/x^3 + 3(2)^2/x^4 + …
(x-2)^-2 = 1/x^2 + 4/x^3 + 12/x^4 + …
Example 2:
Simplify the expression (2x-3)^-2.
Using the (a-b)^-2 formula, we get:
(2x-3)^-2 = 1/(2x)^2 + 2(3)/(2x)^3 + 3(3)^2/(2x)^4 + …
(2x-3)^-2 = 1/4x^2 + 6/4x^3 + 27/4x^4 + …
The (a-b)^-2 formula has numerous applications in mathematics, physics, and engineering. It is used to simplify complex expressions, solve equations, and model real-world problems.
Conclusion
In conclusion, the (a-b)^-2 formula is a powerful tool in mathematics that enables us to simplify complex expressions and solve equations. Its derivation from the binomial theorem and its numerous applications make it an essential concept in algebra. By mastering the (a-b)^-2 formula, students can improve their problem-solving skills and tackle complex mathematical problems with ease.
