Find all sixth roots of the complex number 32√2(1 + i), and plot and label them on thecomplex plane by hand1

Step 1: Convert the complex number to polar form

The complex number 32√2(1 + i) can be written in polar form as r(cos θ + i sin θ), where r is the magnitude of the complex number and θ is the argument.

The magnitude r is given by √((Re(z))^2 + (Im(z))^2) = √((32√2)^2 + (32√2)^2) = 64.

The argument θ is given by arctan(Im(z)/Re(z)) = arctan(1) = π/4.

So, the complex number in polar form is 64(cos(π/4) + i sin(π/4)).

Step 2: Find the sixth roots

The sixth roots of a complex number z = r(cos θ + i sin θ) are given by z_k = r^(1/6)[cos((θ + 2πk)/6) + i sin((θ + 2πk)/6)], for k = 0, 1, ..., 5.

So, the sixth roots of 64(cos(π/4) + i sin(π/4)) are:

z_0 = 2[cos(π/24) + i sin(π/24)], z_1 = 2[cos(5π/24) + i sin(5π/24)], z_2 = 2[cos(9π/24) + i sin(9π/24)], z_3 = 2[cos(13π/24) + i sin(13π/24)], z_4 = 2[cos(17π/24) + i sin(17π/24)], z_5 = 2[cos(21π/24) + i sin(21π/24)].

Step 3: Plot the roots on the complex plane

To plot these roots on the complex plane, you would draw a circle with radius 2 (since the magnitude of all the roots is 2) and then plot the points corresponding to the arguments of the roots. The arguments are π/24, 5π/24, 9π/24, 13π/24, 17π/24, and 21π/24, which are equally spaced around the circle.

Unfortunately, as a text-based AI, I'm unable to draw this by hand. However, you can easily do this on a piece of graph paper or using a graphing tool.