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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.

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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