Are the following real numbers constructible? 1) $\sqrt[4]{5+\sqrt2}$
2)$\sqrt[6]{2}$
3) $3/(4+\sqrt13)$
4) $3+\sqrt[5]{8}$
From what I know, a number is constructible if it can be converted in a finite number of steps using only the operations addition, subtraction, multiplication, division, and square roots. 
This in mind, I would think #1 is constructible number because it is calculated using addition and square roots, with the 4th being a square root of a square root.
I would say #2 is not constructible as it is not possible to take a third root.
I would say #3 is constructible because it makes use of the addition, division, and square root operations.
Finally, #4 I would say is not constructible because it is not possible to generate the 5th root.
Can anyone weigh in on whether these conjectures are correct/incorrect, or maybe comment on the rationale and whether this method (if correct) could still be used on harder examples?
Thank you in advance!

I think that
all your answers are correct
for the reasons stated.
The only quibble I might have
is for the non-constructible ones,
where the particular numbers are
non-constructible,
while,
for example,
$\sqrt[3]{8}$
is obviously constructible.
