Find the two square roots for the following complex number. Write your answers in standard form. (Enter your answers as a comma-separated list.)1 + i3

To find the square roots of a complex number, we first convert the number to polar form.

The complex number 1 + i3 can be written in polar form as r(cos θ + i sin θ), where r is the magnitude of the number and θ is the argument.

1. Calculate r (the magnitude of the complex number): r = sqrt((1)^2 + (3)^2) = sqrt(10)

2. Calculate θ (the argument of the complex number): θ = atan(3/1) = 1.249 radians

So, the polar form of 1 + i3 is sqrt(10)(cos 1.249 + i sin 1.249).

The square roots of a complex number z = r(cos θ + i sin θ) are given by sqrt(r)(cos(θ/2 + kπ) + i sin(θ/2 + kπ)), where k is an integer.

3. Calculate the square roots: For k = 0, we get sqrt(sqrt(10))(cos(1.249/2 + 0π) + i sin(1.249/2 + 0π)) = 1.58 + i0.79 For k = 1, we get sqrt(sqrt(10))(cos(1.249/2 + 1π) + i sin(1.249/2 + 1π)) = -1.58 - i0.79

So, the two square roots of 1 + i3 are 1.58 + i0.79 and -1.58 - i0.79.