Question

The value of i³⁵+i⁹³is equal to

Answer 1

Basic Concept Used :-

$\boxed{ \sf \: {i}^{2} = - 1}$

$\green{\large\underline{\bf{Solution-}}}$

$\red{\bf :\longmapsto\: {i}^{35} + {i}^{93}}$

Consider,

$\blue{\bf :\longmapsto\: {i}^{35}}$

$\rm \: = \: \: i \times {i}^{34}$

$\rm \: = \: \: i \times {({i}^{2})}^{17}$

$\rm \: = \: \: i \times {( - 1)}^{17}$

$\rm \: = \: \: i \times {( - 1)}$

$\blue{\bf \: = \: \: - 1}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \pink{ \boxed{\bf\implies \: {i}^{35} = - 1}}$

Consider,

$\green{\bf:\longmapsto\: {i}^{93}}$

$\rm \: = \: \: i \times {i}^{92}$

$\rm \: = \: \: i \times {( {i}^{2} )}^{46}$

$\rm \: = \: \: i \times {( - 1)}^{46}$

$\rm \: = \: \: i \times 1$

$\green{\bf \: = \: \: 1}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \pink{ \boxed{\bf\implies \: {i}^{93} = 1}}$

Thus,

$\red{\bf :\longmapsto\: {i}^{35} + {i}^{93}}$

$\rm \: = \: \: - 1 + 1$

$\red{\bf \: = \: \: 0}$

Hence,

$\: \: \: \: \: \: \: \: \: \: \: \: \ \: \: \: \: \pink{ \boxed{\bf\implies \: {i}^{35} + {i}^{93} = 0}}$

Additional Information :-

$\boxed{ \sf \:i = \sqrt{ - 1}}$

$\boxed{ \sf \: {i}^{3} = - 1}$

$\boxed{ \sf \: {i}^{4} = 1}$

$\boxed{ \sf \: {i}^{4n} = 1 \: where \: n \in \: natural \: number}$

$\boxed{ \sf \: {i}^{n}+{i}^{n + 1}+{i}^{n + 2}+ {i}^{n + 3}= 0\: where\:n\in\:natural \:number}$

$\boxed{ \sf \:i = - \dfrac{1}{i}}$