De Moivre's theorem:
{r(cosθ + i sinθ)}ⁿ = rⁿ(cos nθ + i sin nθ) (n is an integer)
The argument is multiplied by n, and the absolute value is raised to the nth power.
Example ① Calculate (1+i)¹⁰.
Solution:1+i = √2(cos π/4 + i sin π/4).
(1+i)¹⁰ = 32(cos(5π/2) + i sin(5π/2)) = 32(cos(π/2) + i sin(π/2)) = 32i
Solution:1+i = √2(cos π/4 + i sin π/4).
(1+i)¹⁰ = 32(cos(5π/2) + i sin(5π/2)) = 32(cos(π/2) + i sin(π/2)) = 32i
Key Point
- Using polar form is the shortest route for calculating nth powers.
Practice Problems
- Calculate (√3 + i)⁶.
Solution and Explanation
- Answer:−64