Recognise factors for fractional exponents Hi is there a general rule to factorising these type of exponent fractions? The answer makes sense but I wouldn't have thought of it.
$\ x^{ \frac{2x}{1-2x} } - 2x^{ \frac{1}{1-2x} }$
Answer:
$\ x^{ \frac{2x}{1-2x} } (1- 2x$)

It is a simple algebra: just factor out $x$ to any power you want. For example:
$$\ \color{red}{x^{ \frac{2x}{1-2x} }} - 2x^{ \frac{1}{1-2x} }=\ \color{red}{x^{ \frac{2x}{1-2x} }}\left(1 - 2x^{ \frac{1}{1-2x}-\frac{2x}{1-2x} }\right)=x^{\frac{2x}{1-2x}}\left(1-2x\right) \ \ \text{or}\\
\ x^{ \frac{2x}{1-2x} } - 2\color{blue}{x^{ \frac{1}{1-2x} }}=\ \color{blue}{x^{ \frac{1}{1-2x} }}\left(x^{\frac{2x}{1-2x}-\frac1{1-2x}} - 2\right)=x^{\frac{1}{1-2x}}\left(x^{-1}-2\right)$$
