Find all the complex cube roots of w = 8 (cos 150° + i sin 150°)

Complex numbers are the numbers of the form a + ib, such that a and b are real numbers and i (iota) is the imaginary component and represents √(-1), commonly depicted in their rectangular or standard form. For example, 10 + 5i is a complex number where 10 is the real part and 5i is the imaginary part. These can be purely real or purely imaginary depending upon the values of a and b. If a = 0 in a + ib, then ib is a purely imaginary number, and if b = 0, then we have a, which is a purely real number.

Calculating Roots of Complex Numbers

DeMoivre’s Theorem can be used to simplify higher-order complex numbers. It can be used to determine the roots of complex numbers as well as expand complex numbers according to their exponent.

Given: , then its roots are:

Where, 

k lies between 0 and n – 1 and n is the exponent or radical. 

Find all the complex cube roots of w = 8 (cos 150° + i sin 150°)

Solution:

w = 8(Cos 150° + i sin 150°)

The above complex number can also be expressed as w = 8(cos(150° + 360n) + i sin(150° + 360n).

w^1/3 = {8[(cos(150° + 360n) + i sin(150° + 360n)]}^1/3

= 2^(3)(1/3) [(cos(150° + 360n) + i sin(150° + 360n)]^1/3

As per DeMoivre’s Theorem, (cos x + isinx)ⁿ = cos(nx) + isin(nx). 

= 

Substitute n = 0, 1, 2 to find the roots.

• For n = 0, w₁ =  = 2(cos 50° + i sin 50°)
• For n = 1, w₂ =  = 2(cos 170° + i sin 170°)
• For n = 2, w₃ =  = 2(cos 290° + i sin 290°)

Similar Problems

Question 1. Find all the complex cube roots of w = 125(Cos 150° + i sin 150°). Write the roots in polar form with theta in degrees.

Solution:

w = 125(Cos 150° + i sin 150°)

The above complex number can also be expressed as w = 125 (cos(150° + 360n) + i sin(150° + 360n).

w^1/3 = 

= 5^(3)(1/3) 

As per DeMoivre’s Theorem, (cos x + isinx)ⁿ = cos(nx) + isin(nx). 

= 

Substitute n = 0,1,2 to find the roots.

• For n = 0, w₁ =  = 2(cos 50° + i sin 50°)
• For n = 1, w₂ =  = 2(cos 170° + i sin 170°)
• For n = 2, w₃ =  = 2(cos 290° + i sin 290°)

Question 2. Find all the complex cube roots of w = 27(Cos 150° + i sin 150°). Write the roots in polar form with theta in degrees.

Solution:

w = 27(Cos 150° + i sin 150°)

The above complex number can also be expressed as w = 27(cos(150° + 360n) + i sin(150° + 360n).

w^1/3 = 

= 3^(3)(1/3) 

As per DeMoivre’s Theorem, (cos x + isinx)ⁿ = cos(nx) + isin(nx). 

= 

Substitute n = 0,1,2 to find the roots.

• For n = 0, w₁ =  = 2(cos 50° + i sin 50°)
• For n = 1, w₂ =  = 3(cos 170° + i sin 170°)
• For n = 2, w₃ =  = 3(cos 290° + i sin 290°)

Question 3. Find all the complex cube roots of w = 64 (Cos 150° + i sin 150°). Write the roots in polar form with theta in degrees.

Solution:

w = 64(Cos 150° + i sin 150°)

The above complex number can also be expressed as w = 64(cos(150° + 360n) + i sin(150° + 360n).

w^1/3 = 

= 4^(3)(1/3) 

As per DeMoivre’s Theorem, (cos x + isinx)ⁿ = cos(nx) + isin(nx). 

= 

Substitute n = 0,1,2 to find the roots.

• For n = 0, w1 =  = 4(cos 50° + i sin 50°)
• For n = 1, w2 =  = 4(cos 170° + i sin 170°)
• For n = 2, w3 =  = 4(cos 290° + i sin 290°)

Question 4. Find all the complex cube roots of w = 343 (Cos 150° + i sin 150°). Write the roots in polar form with theta in degrees.

Solution:

w = 343(Cos 150° + i sin 150°)

The above complex number can also be expressed as w = 343(cos(150° + 360n) + i sin(150° + 360n).

w^1/3 = 

= 7^(3)(1/3) 

As per DeMoivre’s Theorem, (cos x + isinx)ⁿ = cos(nx) + isin(nx). 

= 

Substitute n = 0,1,2 to find the roots.

• For n = 0, w1 = = 7(cos 50° + i sin 50°)
• For n = 1, w2 =  = 7(cos 170° + i sin 170°)
• For n = 2, w3 =  = 7(cos 290° + i sin 290°)

Question 5. Find all the complex cube roots of w = 729 (Cos 150° + i sin 150°). Write the roots in polar form with theta in degrees.

Solution:

w = 729(Cos 150° + i sin 150°)

The above complex number can also be expressed as w = 729(cos(150° + 360n) + i sin(150° + 360n).

w^1/3 = 

= 9^(3)(1/3)

As per DeMoivre’s Theorem, (cos x + isinx)ⁿ = cos(nx) + isin(nx). 

= 

Substitute n = 0,1,2 to find the roots.

• For n = 0, w1 = = 9(cos 50° + i sin 50°)
• For n = 1, w2 =  = 9(cos 170° + i sin 170°)
• For n = 2, w3 =  = 9(cos 290° + i sin 290°)