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ayoismeisalex
05.08.2021 •
Mathematics
Find the indicated power using De Moivre’s Theorem (−5−5sqrt 3i)^3
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Ответ:
-5 - 5√3 i = -5 (1 + √3 i )
We have modulus
|-5 (1 + √3 i )| = 5 √(1² + (√3)²) = 5√4 = 10
and argument
arg(-5 - 5√3 i ) = π - arctan(√3) = 2π/3
(we subtract from π because the given complex number lies in the third quadrant of the complex plane, whereas the arctan function only returns angles between -π/2 and π/2)
so that the polar form of the number is
-5 - 5√3 i = 10 exp(2π/3 i )
By DeMoivre's theorem, we have
(-5 - 5√3 i )³ = 10³ exp(3 × 2π/3 i ) = 1000 exp(2πi ) = 1000
Ответ: