Question

evaluate by using complex no (-square root -1)^4n+3

Answer 1

Answer:

$i = \sqrt{-1}$

Step-by-step explanation:

$(-\sqrt{-1} )^{4n+3} \\= (-i)^{4n+3}\\=(-1 * i) ^{4n+3}\\=(-1)^{4n+3} * (i) ^{4n+3}\\=(-1) * i^{4n} * (i)^{3} \\=(-1) * 1 * (-i) \\= i\\=\sqrt{-1}$

Answer 2

There are many methods of solving this question. I have solved it in one of the easiest as well as concise way.

Answer:

$\sqrt{-1}$

Step-by-step explanation:

$(-\sqrt{-1})^{4n+3}$

$= (-i)^{4n+3} \quad [\because \sqrt{-1} = 1]$

$= (-i)^{4n} \times (-i)^3 \quad [\because a^{m+n} = a^m \times a^n]$

$= (i^3)^{4n} \times (-1 \times i)^3 \quad [\because -i  = i^3 \; \; and \; \; -i = -1 \times i]$

$= (i^4)^{3n} \times -1^3 \times i^3$

$= 1^{3n} \times -1 \times -i \quad [\because i^4 = 1 \; \; and \; \; i^3 = -i]$

$= i \quad [\because 1^n = 1]$

$= \sqrt{-1}$

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