1. ## Domain for x^a

For the function $f(x)=5x^{\frac{2}{5}} e^{x}$ , what is the domain of this function?

Mathematica and Matlab says it is only $x\geq 0$ .

But our math teacher did a solution where she looked for the stationary points of $f(x)$ . She answered that $x=- \frac{2}{5}$ was a stationary point. But this value is negative!

She claimed that the domain of $f(x)$ was $\mathbb{R}$ .

2. Does $(-1)^{2/5}$ have a real solution? (I honestly don't know, I REALLY need to read up on my exponents with negative bases), but if a real solution exists for that, then indeed the domain is R.

If only complex solutions exist, the domain is the set of x values such that x is greater than or equal to 0 (assuming the function you're working with maps only to R).

3. Well why not? :-/

Take any negative number,
say -32.

Now (-32)^2/5 is the same as {(-32)^1/5}^2
So basically what we have is 4.

in other words, x^2/5 is merely the square of the fifth root of x. Now the fifth root exists for negative reals as well.

Law of indices. {(a)^b}^c = (a)^(bc)