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.

The magnitude r is given by √(a² + b²), where a and b are the real and imaginary parts of the number, respectively. So, r = √(1² + 3²) = √10.

The argument θ is given by arctan(b/a), so θ = arctan(3/1) = arctan(3).

Now, the square roots of the complex number are given by √r (cos(θ/2 + kπ) + i sin(θ/2 + kπ)), where k is an integer.

For the first root, we take k = 0. So, the first root is √10 (cos(arctan(3)/2) + i sin(arctan(3)/2)).

For the second root, we take k = 1. So, the second root is √10 (cos(arctan(3)/2 + π) + i sin(arctan(3)/2 + π)).

So, the two square roots of 1 + i3 are √10 (cos(arctan(3)/2) + i sin(arctan(3)/2)) and √10 (cos(arctan(3)/2 + π) + i sin(arctan(3)/2 + π)).