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Write sqrt(-16+31i) as a complex number. Can you give me some hints, or if you give me the answer, please tell me the steps.

 Jun 24, 2022
 #1
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In exponential form, -16 + 31i = 34.89*e^(2.04726*pi*i).

 

Taking the square root, we get 5.91*e^(1.024*pi*i).

 

Converting this back to rectangular form gives 2 + 7i.

 Jun 24, 2022
 #2
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Let \(x = \sqrt{-16 + 31i}\) and be in the form a + bi.

 

We have: \((a+bi)^2 = (a+bi)(a+bi) = a^2 + 2abi - b^2 \)

 

We know that \(2abi = 31\), meaning \(ab = {31 \over 2}\)

 

Now, remember that \(a^2 - b^2 = -16\)

 

This gives us the system: 

 

\(a^2 - b^2 = -16\)          (i)

\(ab = {31 \over 2}\)                       (ii)

 

 

Using WA, we find the answer to be: \(\sqrt{-16 + 31i} = \color{brown}\boxed{\sqrt{{\sqrt{1217}\over 2} - 8} + \sqrt{8 + {\sqrt{1217}\over 2}}i}\)

 Jun 24, 2022

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