complex number

Vanni

New member
a cubic root of (-1+i)4√2 is reia with
1) r=2, a=11π/12
2) r=2√2, a=19π/12
3) r=2√2, a=π/4
4) r=2, a=3π/4

I think they are all wrong

Subhotosh Khan

Super Moderator
Staff member
a cubic root of (-1+i)4√2 is reia with
1) r=2, a=11π/12
2) r=2√2, a=19π/12
3) r=2√2, a=π/4
4) r=2, a=3π/4

I think they are all wrong
Is your function (whose cube-root you are seeking):

$$\displaystyle \displaystyle{(-1 + i) * 4 * \sqrt{2} }$$

i.e. you want to evaluate:

$$\displaystyle \displaystyle{[(-1 + i) * 4 * \sqrt{2}]^{\frac{1}{3}} }$$

or something else?

Dr.Peterson

Elite Member
a cubic root of (-1+i)4√2 is reia with
1) r=2, a=11π/12
2) r=2√2, a=19π/12
3) r=2√2, a=π/4
4) r=2, a=3π/4

I think they are all wrong
Possibly you have forgotten that there are three cube roots, and they are asking about one of them, not all or the principal root.

The first answer I come up with is similar to one of the choices, but not identical. However, if I look at the other two roots, one of those is in the list.

Vanni

New member
Possibly you have forgotten that there are three cube roots, and they are asking about one of them, not all or the principal root.

The first answer I come up with is similar to one of the choices, but not identical. However, if I look at the other two roots, one of those is in the list.
the result of my calculations is r = 2 and a = π/4, , I don't understand where I went wrong

Vanni

New member
Is your function (whose cube-root you are seeking):

$$\displaystyle \displaystyle{(-1 + i) * 4 * \sqrt{2} }$$

i.e. you want to evaluate:

$$\displaystyle \displaystyle{[(-1 + i) * 4 * \sqrt{2}]^{\frac{1}{3}} }$$

or something else?
yes
$$\displaystyle \displaystyle{[(-1 + i) * 4 * \sqrt{2}]^{\frac{1}{3}} }$$
this is the one I used for my calculations

Dr.Peterson

Elite Member
the result of my calculations is r = 2 and a = π/4, , I don't understand where I went wrong
You didn't go wrong ... except in stopping here.

Your answer is what I got first. But, as I said, that is only one of three cube roots, and they asked which of the options is a cube root, not which is the one you find first.

Do you know how to find the other roots? You can multiply by each of the roots of unity; or you can just represent the radicand using an alternative angle (a coterminal angle).

Or, you can just cube each of the options and see which works. Note that only two of them have the correct modulus, so that's all you have to try.

pka

Elite Member
yes
$$\displaystyle \displaystyle{[(-1 + i) * 4 * \sqrt{2}]^{\frac{1}{3}} }$$
this is the one I used for my calculations
That number can be written as$$\displaystyle z= -\sqrt8+\sqrt8 i$$. So the polar form is $$\displaystyle z=4\exp\left(\frac{-\pi i}{4}\right)$$
Use $$\displaystyle \rho=\exp\left(\frac{2\pi i}{3}\right)$$ and $$\displaystyle \eta=\sqrt[3]{4}\exp\left(\frac{-\pi i}{12}\right)$$ now the three cube roots of $$\displaystyle z$$ are $$\displaystyle \eta\cdot \rho^k,~k=0,1,2$$

Vanni

New member
You didn't go wrong ... except in stopping here.

Your answer is what I got first. But, as I said, that is only one of three cube roots, and they asked which of the options is a cube root, not which is the one you find first.

Do you know how to find the other roots? You can multiply by each of the roots of unity; or you can just represent the radicand using an alternative angle (a coterminal angle).

Or, you can just cube each of the options and see which works. Note that only two of them have the correct modulus, so that's all you have to try.
So, r=2 and a=11π/12 is correct?

Subhotosh Khan

Super Moderator
Staff member
yes
$$\displaystyle \displaystyle{[(-1 + i) * 4 * \sqrt{2}]^{\frac{1}{3}} }$$
this is the one I used for my calculations
$$\displaystyle \displaystyle{[(-1 + i) * 4 * \sqrt{2}]^{\frac{1}{3}} }$$

= $$\displaystyle \displaystyle{[(cos(3\frac{\pi}{4} \pm 2n\pi) + i sin(3\frac{\pi}{4} \pm 2n\pi) * 8]^{\frac{1}{3}} }$$

= $$\displaystyle \displaystyle{2 * e^{[(cos(\frac{\pi}{4} \pm 2n\pi/3) + i sin(\frac{\pi}{4} \pm 2n\pi/3)]} }$$

Dr.Peterson

Elite Member
So, r=2 and a=11π/12 is correct?
Yes, a could be 3π/12, 11π/12, or 19π/12, corresponding to n=0, 1, 2 in $$\displaystyle {2 e^{\frac{\pi}{4} + \frac{2n\pi}{3}} }$$.

.

pka

Elite Member
That number can be written as$$\displaystyle z= -\sqrt8+\sqrt8 i$$. So the polar form is $$\displaystyle z=4\exp\left(\frac{-\pi i}{4}\right)$$
Use $$\displaystyle \rho=\exp\left(\frac{2\pi i}{3}\right)$$ and $$\displaystyle \eta=\sqrt[3]{4}\exp\left(\frac{-\pi i}{12}\right)$$ now the three cube roots of $$\displaystyle z$$ are $$\displaystyle \eta\cdot \rho^k,~k=0,1,2$$
EDIT
I missread it should be polar form is $$\displaystyle z=4\exp\left(\frac{3\pi i}{4}\right)$$
Use $$\displaystyle \rho=\exp\left(\frac{2\pi i}{3}\right)$$ and $$\displaystyle \eta=\sqrt[3]{4}\exp\left(\frac{3\pi i}{12}\right)$$ now the three cube roots of $$\displaystyle z$$ are $$\displaystyle \eta\cdot \rho^k,~k=0,1,2$$