Factoring Quantities Question I was doing an exercise and ran into a problem with their use of factoring. Here is the problem specific to where the issue occurs:
$$
\frac{(x^2 + 1)^{1/2} - x^2 (x^2 + 1)^{-1/2}}{x^2 + 1} = \frac{(x^2 + 1)^{-1/2}\left( x^2 + 1 - x^2\right)}{x^2 + 1}
$$
If $(x^2+1)^{-1/2}$ was factored out on the top, shouldn't the amount after factoring be just be $1 - x^2$? I am not seeing where the extra $x^2$ is coming from. Can anyone spot the issue I am having?

When you factor out a $(x^2+1)^{-\frac{1}{2}}$, the remaining quantity is multiplied by $(x^2+1)^{\frac{1}{2}}$. That is, if we have some expression
$$(x^2+1)^a \cdot f(x)$$
where $f(x)$ is some polynomial and $a$ is any number, when we factor out a $(x^2+1)^{-\frac{1}{2}}$ we get
$$(x^2+1)^{-\frac{1}{2}}\left((x^2+1)^{a+\frac{1}{2}}\cdot f(x)\right).$$
More concretely in the case of your example, because $\frac{1}{2}=-\frac{1}{2}+1$, we have that
$$(x^2+1)^{\frac{1}{2}}=(x^2+1)^{-\frac{1}{2}}(x^2+1)^1,$$
and therefore
$$\begin{align} (x^2+1)^{\frac{1}{2}}-x^2(x^2+1)^{-\frac{1}{2}} =&(x^2+1)^{-\frac{1}{2}}(x^2+1)^1-x^2(x^2+1)^{-\frac{1}{2}}\\ =& (x^2+1)^{-\frac{1}{2}}\left((x^2+1)-x^2\right)\end{align}$$
