<s>[INST] 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:
$\ x^{ \frac{2x}{1-2x} } (1- 2x$)
 [/INST] 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}
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