Finding a complex number So I have an illustrative picture of all of the roots of one complex number (the exponent is random). I have to find this number $z$, if we
know that these roots are on a circle, that's diameter is $3.4$ and degree between the real axis and the positive direction is $-61$ degrees.

Here is the illustrative picture that I have:



I got the answer that the complex number is $z=0.824+1.487i$.
but all I really need is $a$

Are my calculations right?
Assuming I understand the question correctly, we have:
$$w=z^{\frac{1}{6}}=\frac{3.4}{2} \cdot (\cos{(-61)}+i \cdot  \sin{(-61)}) = 0.82418-1.48685i$$
Which is what you have. But this is $w=z^{\frac{1}{6}}$ ;
$$\therefore z = w^6 = 24.00534-2.52306i$$
Change the $+$ to a $-$ and you're golden.

Presumably you have calculated
$$x = 1.7\cos(-61^\circ), y = 1.7\sin(-61^\circ),$$ which does indeed give the answer you suggest.