Find x and y in complex numbers

Question

Find x and y in complex numbers

(X+yi)⁴= -7-12i

 

 

 

Answer 1

Mustafa,

 

Basically what you need here are the four 4th roots of the complex number −7 − 12i.

 

The steps to do this are as follows:

 

1) Write the number in trigonometric form: −7 − 12i = √193 cis 192.18°.

2) Use the nth Root Theorem to find all four roots:

 

w_k = (√193)^1/4 cis (192.18°/4 + 360°k/4) for k = 0, 1, 2, 3

 

After plugging in and simplifying you will get 

 

w₀ = ⁸√193 cis 48.04°

w₁ = ⁸√193 cis 138.04°

w₂ = ⁸√193 cis 228.04°

w₃ = ⁸√193 cis 318.04°

 

You can then use the definition of cis θ = cos θ + i sin θ to write these four complex numbers in standard form x + yi. You will need to do this using your own calculator. As you did not specify how the answers were to be given, I cannot finish the problem for you. The x and y values of the answers will probably all involve decimals. You must round them as requested in the original problem.

Answer 2

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