Finding the Square Root of 9 + 40i

To find the square root of a complex number, we can use the polar form of the number. The polar form of a complex number is expressed as r(cosθ + isinθ), where r is the modulus (magnitude) and θ is the argument (angle).

Step 1: Convert the Complex Number to Polar Form
We need to convert the complex number 9 + 40i into polar form. To do this, we can use the following formulas:

r = √(a² + b²)
θ = arctan(b/a)

For 9 + 40i:
a = 9
b = 40

Calculating the modulus (r) and argument (θ), we have:
r = √(9² + 40²) = √(81 + 1600) = √1681 = 41
θ = arctan(40/9) ≈ 77.47°

So, the polar form of 9 + 40i is 41(cos77.47° + isin77.47°).

Step 2: Finding the Square Root
To find the square root of a complex number in polar form, we can use the following formula:

√(r(cosθ + isinθ)) = √r(cos(θ/2) + isin(θ/2))

Using this formula, we can find the square root of 41(cos77.47° + isin77.47°):

√(41(cos77.47° + isin77.47°)) = √41(cos(77.47°/2) + isin(77.47°/2))

Simplifying further:
√41(cos38.74° + isin38.74°)

Step 3: Convert the Square Root to Rectangular Form
To convert the square root back to rectangular form, we can use the following formulas:

a = rcosθ
b = rsinθ

For √41(cos38.74° + isin38.74°):
a = √41cos38.74° ≈ 5.899
b = √41sin38.74° ≈ 3.667

So, the square root of 9 + 40i is approximately 5.899 + 3.667i.

Summary:
- Convert the complex number to polar form using the modulus and argument formulas.
- Apply the square root formula to the polar form.
- Convert the square root back to rectangular form using the cosine and sine formulas.
- The final result is the square root of the complex number in rectangular form.