Question

Solve the following expression using De Moivre's Theorem:

(2 + 3i)⁴

a) 16 + 96i
b) -104 + 48i
c) -104 - 48i
d) 16 - 96i

Answer 1

To solve (2 + 3i)⁴ using De Moivre's Theorem, convert the complex number to polar form, apply the theorem, and then convert back to rectangular form.

Step-by-step explanation:

The question asks to solve the expression (2 + 3i)⁴ using De Moivre's Theorem. To do this, we first need to convert the complex number 2 + 3i into polar form, which is r(cosθ + i sinθ), where r is the magnitude of the complex number and θ is the argument.

First, we find r using the formula r = √(a² + b²) where a = 2 and b = 3. So, r = √(2² + 3²) = √(4 + 9) = √13.

Next, we find θ which is the arctan(b/a), so θ = arctan(3/2).

Now we apply De Moivre's Theorem, which states that (r(cosθ + i sinθ))⁴ = r⁴(cos(4θ) + i sin(4θ)). Therefore, (2 + 3i)⁴ = (√13⁴)(cos(4 arctan(3/2)) + i sin(4 arctan(3/2))).

After calculating cos(4 arctan(3/2)) and sin(4 arctan(3/2)), the results can be multiplied by r⁴, which is 13² to get the final complex number in rectangular form (a + bi), where a is the real part and b is the imaginary part.