Question

Solve : $\left(\frac{-1-i}{\sqrt{2}}\right)^{100}$​

Answer 1

$\huge \boxed{ \underline{ \underline{ \bf{Answer}}}}$

$\sf{ \: To \: Solve : ( {\frac{ - 1 - i }{ \sqrt{2} } )}^{100} }$

SOLUTION :

$= \sf {\: ( {\frac{ - 1 - i }{ \sqrt{2} } )}^{100}}$

$= \: \sf{ { [( {\frac{ - 1 - i }{ \sqrt{2} } )}^{2} ]}^{50} }$

$= \: \sf{{ [{\frac{ (- 1 - i) }{ ({ \sqrt{2} )}^{2}}}^{2} ]}^{50}}$

$= \: \sf{ { [{\frac{ ({ - 1)}^{2} + \: {i} \: ^{2} - ( - 2i) }{2 }} ]}^{50}}$

$= \: \sf{ { [\frac{1 \: + \: ( - 1) \: + 2i}{2} ]}^{50} }$

$= \: \sf{ {( \frac{2i}{2} )}^{50} }$

$= \: \sf{{(i)}^{50} }$

We can write it like this,

$= \: \sf {{ ({i}^{4})}^{12} . \: {i}^{2} }$

$= \: \sf{ {1}^{12} . {i}^{2} }$

$\: = \sf{\: {i}^{2} }$

$= \: - 1$

HENCE OUR REQUIRED ANSWER IS -1.

Answer 2

Answer:

Step-by-step explanation:

Complex numbers:-

We are provided with a complex number.

We know,cos45°=sin45°=1/✓2

We convert the given complex number into polar form.

Then we apply De Moivre's theorem.

De Moivre's theorem:-

(cosø+isinø)^n=cosnø+isinnø

cos25π=-1,sin25π=0

Hence, answer is -1