Find the indicated roots of the following. Express your answer in the form found using Euler's Formula, |z|neinθ.
The cube roots of −4+i
-4+i = √17 cis tan-1(-1/4) = 4.1231 cis 2.8966
so the 4th roots are
1.60352 cis 2.8966/3 = 1.60352 cis 0.9655 + k * 2π/3
I got
z^(1/4) = (√17)^(1/4) cis (2.8966/4)
primary root = 1.425 cis .7242
general solution:
1.425 cis (.7242 + k*π/2) where k = 0,1,2,3
(you are dividing 2π into 4 parts , not 3
you have to take the 4th root of √17, you took the third)
in degrees
1.425 cis (41.49° + k(90) ), for k = 0,1,2,3
??
The cube roots of −4+i
I think I should have read the original question except the second line of your answer, where you are talking about the fourth root.
Sorry about the confusion.
well, I'm sure by now @Anah will have sorted it all out ...
To find the cube roots of a complex number, we can use Euler's formula, which states that any complex number can be written as:
z = |z|e^(iθ)
where |z| is the magnitude of the complex number and θ is the argument of the complex number.
Let's start by finding the magnitude and argument of the complex number -4+i.
Magnitude (|z|):
The magnitude of a complex number is given by the formula:
|z| = sqrt(Re(z)^2 + Im(z)^2)
For -4+i, the real part (Re(z)) is -4 and the imaginary part (Im(z)) is 1. So,
|z| = sqrt((-4)^2 + 1^2) = sqrt(17)
Argument (θ):
The argument of a complex number can be found using the inverse tangent function:
θ = arctan(Im(z) / Re(z))
For -4+i, Re(z) is -4 and Im(z) is 1. So,
θ = arctan(1 / -4) = arctan(-0.25)
Now we have the magnitude and argument of -4+i. Let's express the cube roots of -4+i in the form found using Euler's formula, |z|ne^(iθ).
Cube roots:
To find the cube root of a complex number, we can take its magnitude to the power of 1/3 and multiply the argument by 1/3. So, to find the cube roots of -4+i, we have:
Cube root 1: |z|^(1/3)e^(iθ/3)
= (sqrt(17))^(1/3)e^(i(arctan(-0.25)/3))
Cube root 2: |z|^(1/3)e^(i(θ/3 + 2π/3))
= (sqrt(17))^(1/3)e^(i((arctan(-0.25)/3) + 2π/3))
Cube root 3: |z|^(1/3)e^(i(θ/3 + 4π/3))
= (sqrt(17))^(1/3)e^(i((arctan(-0.25)/3) + 4π/3))
So, the cube roots of -4+i, expressed in the form found using Euler's formula, |z|ne^(iθ), are:
Cube root 1: (sqrt(17))^(1/3)e^(i(arctan(-0.25)/3))
Cube root 2: (sqrt(17))^(1/3)e^(i((arctan(-0.25)/3) + 2π/3))
Cube root 3: (sqrt(17))^(1/3)e^(i((arctan(-0.25)/3) + 4π/3))