SOLUTION: How do i solve the equation _ What is the square root of 4-7i?

Question 397698: How do i solve the equation _ What is the square root of 4-7i?
Found 3 solutions by Alan3354, robertb, richard1234:
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
That's not an equation, there's no equal sign.
Use DeMoivre's Theorem to find the sq root.

Answer by robertb(5830)   (Show Source): You can put this solution on YOUR website!
Suggestion:
Let , so that .
==> , and . After substitution,
, or .
==> . Eliminate the negative value to get
.
Get the square root of this, and plug into .
Good luck!

Answer by richard1234(7193)   (Show Source): You can put this solution on YOUR website!
It is indeed possible to solve it using Euler's formula, by multiplying/dividing by to obtain a unit vector and obtaining:

where and .

However, the easiest way is probably to assume that the square root of is for real numbers a,b. Squaring both sides,

Equating the real and imaginary parts, we get the system of equations

--> ab = -7/2.

Using the substitution b = -7/2a,

Use the quadratic formula to solve for then take the square root. Note that must be positive based on our assumption that a and b are real numbers. After that, the rest is just arithmetic.
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