(a) We know `root3(1+x) = (1+x)^(1//3)`.

Substituting `n=1/3` into the binomial series, we get:


`~~ 1+1/3 x` `+((1/3)(1/3-1))/2 x^2` `+ ((1/3)(1/3-1)(1/3-2))/6 x^3+...`

`= 1+x/3-(x^2)/9+(5x^3)/81-...`

(b) We now use the above expression to approximate `root3(1.2)`.

Substituting `x=0.2` into our expansion for `root3(1+x)`, we have:

`root3(1.2)` `~~ 1+(0.2)/3` `-((0.2)^2)/9` `+(5((0.2)^3))/81-...` `=1.06271604938`

Is it correct?

The calculator value for `root3(1.2)` is `1.06265856918`, so our approximation using binomial series is accurate to the 4th decimal place. Taking more terms of the series would give us a more accurate result.