how to calculate 2^1.4

You can use Newtons approximation method.

We want to find $x=2^{1.4}$, or equivalently, $x^5=\left(2^{1.4}\right)^5=2^7=128$

Define $f(x)=x^5-128$

We want to find the root of $f(x)$

as noted in my comment, $x\approx \sqrt{8} \approx 2.828$

So we start with this guess of $2.828$.

We put it into the formula,

$$x'=x-\frac{f(x)}{f'(x)}$$

and $x'$ will be a more accurate guess.

$$x'= 2.828 - \frac{(2.828)^5-128}{5(2.828)^4}$$

with a bit of hand calculation we get $x'=2.66\ldots$ which is very close to the actual value.

$$2^{1.4}=e^{1.4\ln 2}=\sum_{k=0}^\infty\frac{(1.4\ln 2)^k}{k!}$$