$$\begin{aligned} &\frac{(\frac{1}{x^2}+x)(\frac{1}{x^2}-x)}{1-x^6} \\ &= \frac{(\frac{1}{x^2}+x)(\frac{1}{x^2}-x)}{1-x^6} (\frac{x^2}{x^2}) \\ &= \frac{(1+x^3)(1-x^3)}{x^2(1-x^6)} \\ &= \frac{1-x^6}{x^2(1-x^6)} \\ &= \frac{1}{x^2} \end{aligned} $$
Hello, can somebody help me understand why this method doesn't give the correct answer? The correct answer is $\frac{1}{x^4}$ and I am able to produce the correct answer if I multiply the two $x$'s in the numerator by $\frac{x^2}{x^2}$ instead of multiplying the big overall fraction by $\frac{x^2}{x^2}$ as shown above. I also know that I am able to get the correct answer if I FOIL out the numerator before multiplying by $\frac{x^4}{x^4}$ (since the new denominator in the numerator becomes $x^4$), but why can I not get the right answer if I multiply by $\frac{x^2}{x^2}$ right off the bat?
The problem is that, when you distribute the $x^2$ into both the numerator and the denominator, you add an $x^2$ in each factor on the numerator. You are essentially multiplying the numerator by $x^4$ but the denominator by $x^2$. This is why your final answer is off by a factor of $x^2$.