Use de Moivre's Theorem to find the following. Write your answer in standard form.(3 − i)4

Sure, let's use De Moivre's Theorem to solve this problem. De Moivre's Theorem states that for any real number x and integer n, (cos x + i sin x)^n = cos nx + i sin nx.

Step 1: Convert (3 - i) to polar form.

The magnitude r is sqrt(3^2 + (-1)^2) = sqrt(10).

The argument θ is arctan(-1/3) = -0.32175 (in radians).

So, (3 - i) = sqrt(10) * (cos(-0.32175) + i sin(-0.32175)).

Step 2: Apply De Moivre's Theorem.

(sqrt(10) * (cos(-0.32175) + i sin(-0.32175)))^4 = (sqrt(10))^4 * (cos(4*-0.32175) + i sin(4*-0.32175)).

Step 3: Simplify the expression.

This equals 100 * (cos(-1.287) + i sin(-1.287)).

Step 4: Convert back to rectangular form.

This equals 100 * (0.2756 - 0.9613i) = 27.56 - 96.13i.

So, (3 - i)^4 = 27.56 - 96.13i.