%5E-2-binomial-expansion.jpeg "(a+b)^-2 Binomial Expansion")
Binomial Expansion: $(a+b)^{-2}$
In algebra, binomial expansion is a useful technique for expanding powers of a binomial expression, which is an expression consisting of two terms. In this article, we will explore the binomial expansion of $(a+b)^{-2}$.
What is Binomial Expansion?
Binomial expansion is a method of expanding a binomial expression raised to a power. It is based on the formula:
$(a+b)^n=\sum_{k=0}^{n}{n\choose k}a^{n-k}b^k$
where $n$ is a positive integer, and ${n\choose k}$ is the binomial coefficient, which can be calculated as:
${n\choose k}=\frac{n!}{k!(n-k)!}$
Binomial Expansion of $(a+b)^{-2}$
Now, let's apply the binomial expansion formula to $(a+b)^{-2}$. Since $n=-2$, we can write:
$(a+b)^{-2}=\sum_{k=0}^{-2}{{-2}\choose k}a^{-2-k}b^k$
Using the formula for binomial coefficient, we can calculate the values of ${-2\choose k}$:
${-2\choose 0}=\frac{(-2)!}{0!(-2)!}=1$
${-2\choose 1}=\frac{(-2)!}{1!(-3)!}=-2$
${-2\choose 2}=\frac{(-2)!}{2!(-4)!}=1$
Substituting these values into the expansion, we get:
$(a+b)^{-2}=a^2-2ab+b^2$
Simplification
We can simplify the expansion by combining like terms:
$(a+b)^{-2}=a^2-2ab+b^2=\frac{1}{a^2-2ab+b^2}$
Conclusion
In this article, we have derived the binomial expansion of $(a+b)^{-2}$. We have seen how to apply the binomial expansion formula to obtain the expansion, and then simplify it to obtain a more compact form. This expansion is useful in many algebraic manipulations and has numerous applications in mathematics, physics, and engineering.
