find-the-first-root-8i-1-2-

Question Number 139608 by mohammad17 last updated on 29/Apr/21

f i n d t h e f i r s t r o o t {(- 8 i)}^{\frac{1}{2}}

Answered by floor(10²Eta[1]) last updated on 29/Apr/21

\sqrt{- 8 i} = a + bi

- 8 i = a^{2} - b^{2} + 2 abi

\Rightarrow a^{2} - b^{2} = 0 \land - 8 = 2 ab

∣ a ∣=∣ b ∣ \land ab = - 4

b^{2} = 4 \Rightarrow b = \pm 2, a = \pm 2

\sqrt{- 8 i} = 2 - 2 i or - 2 + 2 i

Answered by mr W last updated on 29/Apr/21

- 8 i = 8 e^{(2 k π - \frac{π}{2}) i}

{(- 8 i)}^{\frac{1}{2}} = 2 \sqrt{2} e^{(k - \frac{1}{4}) π i}, k = 0, 1

= 2 \sqrt{2} (\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}} i) = 2 - 2 i

o r = 2 \sqrt{2} (- \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} i) = - 2 + 2 i

Facebook Tweet Pin