Find (-1+i)^1/3 list steps
At what level are you studying this?
Do you know DeMoivre's Theorem ?
change to polar form
r(cosØ + isinØ)
r = √(1+1) = √2
tan^-1 Ø = -1 (in II)
Ø = 135°
-1 + i = √2(cos135° + i sin135°)
(-1+i)^(1/3) = √2^(1/3)(cos 135/3 + i sin 135/3)
= 2^(1/6)(cos 45° + i sin 45°)
= 2^(1/6)(√2/2 + i√2/2)
= ( 2^(2/3) + i 2^(2/3) )/2
= 2^(-1/3) + i 2^(-1/3) or appr. .7937 + i(.7937)
check my arithmetic
To find the cube root of a complex number, such as (-1+i)^(1/3), we can follow these steps:
Step 1: Rewrite the complex number in polar form.
To express (-1+i) in polar form, we need to determine the magnitude (r) and argument (θ) of the complex number. The magnitude (r) can be found using the formula |z| = √(Re^2 + Im^2), where Re is the real part and Im is the imaginary part. For (-1+i), Re = -1 and Im = 1, so we have |z| = √((-1)^2 + 1^2) = √(1 + 1) = √2.
To find the argument (θ), we use the formula θ = arctan(Im/Re), taking into account the quadrant in which the complex number lies. In the case of (-1+i), it lies in the second quadrant, so we have θ = arctan(1/(-1)) = arctan(-1) = -π/4.
Therefore, (-1+i) can be expressed as r * e^(iθ), where r = √2 and θ = -π/4.
Step 2: Express the cube root of (-1+i) in polar form.
To find the cube root of a complex number in polar form, we need to take the cube root of the magnitude (r) and divide the argument (θ) by 3. In this case, the cube root of √2 is (√2)^(1/3).
For the argument, we divide -π/4 by 3 to get -π/12.
Hence, the cube root of (-1+i) in polar form is (√2)^(1/3) * e^(i(-π/12)).
Step 3: Convert the result back to rectangular form.
To convert the result in polar form back to rectangular form, we can use Euler's formula, which states that e^(iθ) = cos(θ) + i*sin(θ).
So, (√2)^(1/3) * e^(i(-π/12)) can be written as (√2)^(1/3) * (cos(-π/12) + i*sin(-π/12)).
Evaluating cos(-π/12) and sin(-π/12) will give us the real and imaginary parts, respectively.
Step 4: Simplify the expression to get the final result.
By calculating (√2)^(1/3) * (cos(-π/12) + i*sin(-π/12)), we obtain:
(√2)^(1/3) * (cos(-π/12) + i*sin(-π/12)) = (√2)^(1/3) * (cos(-π/12)) + (√2)^(1/3) * (i*sin(-π/12)).
From here, you can evaluate the expressions using a calculator or leave the answer as (√2)^(1/3) * (cos(-π/12)) + (√2)^(1/3) * (i*sin(-π/12)), which represents the cube root of (-1+i) in rectangular form.