Question

Calculate the expression (1/2 + i√3/2)^5.

Answer 1

The expression (1/2 + i√3/2)
`^5` represents the fifth power of the cosine and sine representations of an angle, which is solved by using De Moivre's Theorem. The final answer is 1/2 - i√3/2.

Step-by-step explanation:

The expression (1/2 + i√3/2)
`^5` can be recognized as the fifth power of the complex number representation of the cosine and sine of an angle in polar form, particularly the formula for cos(60°) + isin(60°). This is a well-known angle with a reference on the unit circle at 60 degrees, and taking it to the fifth power involves using De Moivre's Theorem, which states that for a complex number in polar form (r(cos θ + i sin θ))
`^n` = r
`^n` (cos(nθ) + i sin(nθ)). Applying this theorem, we multiply the angle by 5.

Since the radius (r) is 1 in this case, we have:

(1(cos 60° + i sin 60°))
`^5` = cos(300°) + i sin(300°),
which is equivalent to:
cos(-60°) + i sin(-60°), since cosine is even and sine is odd function.

Therefore, we end up with the final result:
cos(-60°) + i sin(-60°) = 1/2 - i√3/2.