Recognise factors for fractional exponents

Question

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: $\ x^{ \frac{2x}{1-2x} } (1- 2x$)

Answer 1

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

Answer 2

This has more to do with the manipulation of rational expressions.

We have $$\frac1{1-2x}=\frac{1\color{blue}{-2x+2x}}{1-2x}=1+\frac{2x}{1-2x}.$$

As such, \begin{align} x^{\frac{2x}{1-2x}}-2x^{\frac1{1-2x}}&=x^{\frac{2x}{1-2x}}-2x^{1+\frac{2x}{1-2x}}\\ &=x^{\frac{2x}{1-2x}}-2x\cdot x^{\frac{2x}{1-2x}}\\ &=x^{\frac{2x}{1-2x}}(1-2x). \end{align}