Use De Moivre’s Theorem to simplify each expression. Write the answer in the form a + bi.
{1 - i(/3)}^4
I will assume you know the theorem and the common variables used in it
r = √(1^2 + (1/3)^2 )
= √(1 + 1/9)
= √(10/9)
√10/3
let the angle be Ø
cosØ = 1/(√10/3) = 3/√10
sinØ = (-1/3) / (√10/3) = -3/√10 , so Ø is in IV and Ø = 341.565°
so (1 - (1/3) i )^4 = (√10/3)^4 [ cos 4(341.565° + i sin 4(341.564°)
= (100/81) [ .28 + (-.96i) ]
= 28/81 - 96/81i
= 28/81 - 32/27 i
=
To simplify the expression using De Moivre's Theorem, we first rewrite the expression in polar form.
Polar form of a complex number z = r(cosθ + isinθ), where r is the magnitude of z and θ is the argument of z.
To express the complex number 1 - i/√3 in polar form, we need to find its magnitude and argument.
Magnitude (r):
The magnitude of a complex number z = x + yi is given by r = sqrt(x^2 + y^2).
For 1 - i/√3, the magnitude is:
r = sqrt((1)^2 + (-1/√3)^2)
= sqrt(1 + 1/3)
= sqrt(4/3)
= 2/sqrt(3)
Argument (θ):
The argument of a complex number z = x + yi is given by θ = atan(y/x).
For 1 - i/√3, the argument is:
θ = atan((-1/√3)/1)
= atan(-1/√3)
= -30° (approximately)
Now, we can express 1 - i/√3 in polar form as:
1 - i/√3 = (2/√3)(cos(-30°) + isin(-30°))
De Moivre's Theorem states:
For any complex number z = r(cosθ + isinθ), and any positive integer n, the nth power of z can be expressed as:
z^n = r^n(cos(nθ) + isin(nθ))
Using De Moivre's Theorem, we can simplify the expression (1 - i/√3)^4.
(1 - i/√3)^4 = (2/√3)^4 (cos(-30°*4) + isin(-30°*4))
= (2/√3)^4 (cos(-120°) + isin(-120°))
Now, let's evaluate the expression:
(2/√3)^4 = (2^4)/(√3)^4
= 16/9
cos(-120°) = -1/2
sin(-120°) = -sqrt(3)/2
Therefore, the simplified expression is:
(1 - i/√3)^4 = (16/9)(-1/2 - sqrt(3)/2i)
= -8/9 - 8sqrt(3)/9i
So, the answer in the form of a + bi is:
-8/9 - 8sqrt(3)/9i.