Find the square root of 5-12i

To find the square root of 5 - 12i, we can use the formula for the square root of a complex number.

Let's express 5 - 12i in polar form.

The modulus or absolute value of a complex number is given by:
|r| = √(a^2 + b^2), where a is the real part and b is the imaginary part.

|r| = √(5^2 + (-12)^2) = √(25 + 144) = √169 = 13

The argument or angle of a complex number is given by:
θ = arctan(b/a), where a is the real part and b is the imaginary part.

θ = arctan(-12/5) ≈ -67.38°

Thus, 5 - 12i can be expressed in polar form as 13(cos(-67.38°) + sin(-67.38°)).

Now, for the square root of this number, we need to find another complex number whose square is equal to this number.

The square root formula for a complex number in polar form is:
√(r(cosθ + sinθ)) = √r(cos(θ/2) + sin(θ/2))

Applying this formula, we have:
√(13(cos(-67.38°) + sin(-67.38°))) = √13(cos(-67.38°/2) + sin(-67.38°/2))

√(13(cos(-67.38°) + sin(-67.38°))) ≈ √13(cos(-67.38°/2) + sin(-67.38°/2))

Using a calculator, we can evaluate this approximation:

√(13(cos(-67.38°/2) + sin(-67.38°/2))) ≈ √13(0.686 + 0.728i)

Therefore, the square root of 5 - 12i is approximately 0.686 + 0.728i.