Answer:
Valid only for |x|<1
Step-by-step explanation:
Binomial expansion for rational powers is valid only if
|x|<1
If |x|<1 we have
[tex](1-x)^{\frac{1}{5} } =1+\frac{1}{5} (-x)+\frac{\frac{1}{5} \frac{-4}{5}x^2}{2!} +\frac{\frac{1}{5} \frac{-4}{5} \frac{-9}{5}x^3 }{3!} +...[/tex]
Same like integral powers except that instead of nCr we write here n(n-1)../r!
and there will be an infinite series
Thus we have
[tex](1-x)^{\frac{1}{5} } =1+\frac{1}{5} (-x)+\frac{\frac{1}{5} \frac{-4}{5}x^2}{2!} +\frac{\frac{1}{5} \frac{-4}{5} \frac{-9}{5}x^3 }{3!} +..\\= 1-\frac{x}{5} -\frac{2x^2}{25} +\frac{12x^3}{125} +...[/tex]
