Questions: Raise (1+sqrt3 i) to the (4^th) power and express the answer in rectangular form. Select one: a. (-8 sqrt3+8 i) b. (-8-8 sqrt3 i) c. (8+8 sqrt3 i) d. (-128-128 sqrt3 i)

Solution Steps

To solve this problem, we need to raise the complex number \(1+\sqrt{3}i\) to the 4th power. We can use the binomial theorem or directly use Python's complex number arithmetic to compute this. The result will be expressed in rectangular form, which is \(a + bi\).

The given complex number is \( z = 1 + \sqrt{3}i \).

We need to compute \( z^4 \). Using the properties of complex numbers, we calculate: \[ z^4 = (1 + \sqrt{3}i)^4 \]

After performing the calculation, the result is: \[ z^4 = -8 - 13.8564i \]

The rectangular form of the result is already given as: \[ -8 - 13.8564i \]

Comparing the calculated result with the provided options, none of the options exactly match the calculated result. However, the closest option in terms of structure is: b. \(-8 - 8\sqrt{3}i\)

Final Answer