Question

Finding a Power of a Complex Number: Use DeMoivre's Theorem to find (1+i√3)⁶*².

a) 1
b) 1 + i√3
c) 1 - i√3
d) 2

Answer 1

To find the power of a complex number using DeMoivre's Theorem, we need to express the complex number in polar form. In this case, (1+i√3)⁶*² simplifies to 64.

Step-by-step explanation:

To find the power of a complex number using DeMoivre's Theorem, we need to express the complex number in polar form.

To do this, we can write (1+i√3) as r(cosθ + isinθ), where r is the magnitude of the complex number and θ is the argument.

In this case, the magnitude r is √((1)^2 + (√3)^2) = 2, and the argument θ is arctan(√3/1) = π/3.

Using DeMoivre's Theorem, we can then raise the complex number to the power of 6*² by taking the magnitude to the power of 6*² and multiplying the argument by 6*².

Therefore, (1+i√3)⁶*² = 2⁶*² × (cos(6*²π/3) + isin(6*²π/3)).

Simplifying further, we have (1+i√3)⁶*² = 2⁶*² × (cos(4π) + isin(4π)).

Since cos(4π) = cos(0) = 1 and sin(4π) = sin(0) = 0, the complex number simplifies to 2⁶*² × (1 + i(0)).

Finally, 2⁶*² × (1 + i(0)) = 2⁶*² = 64.