Question

How do I expand this binomial?

\[(x^2 + y^2)^{\frac{1}{2}} = x(1 + \frac{y^2}{x^2})^{\frac{1}{2}} = x(1 + \frac{y^2}{2x^2} + \frac{y^4}{x^4})\]

I understand how you get \[(x^2 + y^2)^{\frac{1}{2}} = x(1 + \frac{y^2}{x^2})^{\frac{1}{2}}\] but not how you get \[x(1 + \frac{y^2}{2x^2} + \frac{y^4}{x^4})\].

My first question is: how do you know the first step is to simplify \[(x^2 + y^2)^{\frac{1}{2}}\] to \[x(1 + \frac{y^2}{x^2})^{\frac{1}{2}}\]?

My second question is how to expand a binomial with a fraction power.

Answer 1

for the first question, you just factor x^2 out and then take it out of the sqrt. like
\[\sqrt{x^2(1+ \frac{ y^2 }{ x^2 })}= x \sqrt{1+\frac{ y^2 }{ x^2 }}=x(1+ \frac{ y^2 }{ x^2 })\]

Answer 2

the last term ^1/2

Answer 3

to open the binomial aply binomial theorem\[(a+b)^m = \sum_{k=0}^{\infty}\left(\begin{matrix}m \\ k\end{matrix}\right)a^{m-k}b^k\]

Answer 4

your m is 1/2 your k is 2 and a =1, b = y^2/x^2