Question

Simplify =>
i⁵⁹²+i⁵⁹⁰+i⁵⁸⁸+i⁵⁸⁶+i⁵⁸⁴/i⁵⁸²+i⁵⁸⁰+i⁵⁷⁸+i⁵⁷⁶+i⁵⁷⁴​​

Answer 1

Step-by-step explanation:

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Answer 2

$\large\underline{\sf{Solution-}}$

Given expression is

$\rm \: \dfrac{{i}^{592} + {i}^{590} + {i}^{588} + {i}^{586} + {i}^{584}}{{i}^{582} + {i}^{580} + {i}^{578} + {i}^{576} + {i}^{574}}$

$\rm \: = \: \dfrac{{i}^{584}({i}^{8} + {i}^{6} + {1}^{4} + {i}^{2} + 1)}{{i}^{574}({i}^{8} + {i}^{6} + {1}^{4} + {i}^{2} + 1)}$

$\rm \: = \: \dfrac{{i}^{584}}{{i}^{574}}$

We know,

$\boxed{ \rm{ \: {x}^{m} \div {x}^{n} \: = \: {x}^{m - n} \: \: }}$

So, using this identity, we get

$\rm \: = \: {i}^{584 - 574}$

$\rm \: = \: {i}^{10}$

$\rm \: = \: {( {i}^{2}) }^{5}$

We know,

$\color{green}\boxed{ \rm{ \: \: {i}^{2} \: = \: - \: 1 \: \: }}$

So, using this result, we get

$\rm \: = \: {( - 1)}^{5}$

$\rm \: = \: - 1$

Hence,

$\color{green}\boxed{ \rm{ \:\rm \: \dfrac{{i}^{592} + {i}^{590} + {i}^{588} + {i}^{586} + {i}^{584}}{{i}^{582} + {i}^{580} + {i}^{578} + {i}^{576} + {i}^{574}} \: = \: - \: 1 \: }}$

$\rule{190pt}{2pt}$

Additional Information :-

Argument of complex number :-

$\color{blue}\begin{gathered}\boxed{\begin{array}{c|c} \bf Complex \: number & \bf arg(z) \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf x + iy & \sf  {tan}^{ - 1}\bigg |\dfrac{y}{x} \bigg|   \\ \\ \sf  - x + iy & \sf \pi - {tan}^{ - 1}\bigg |\dfrac{y}{x}\bigg | \\ \\ \sf  - x - iy & \sf  - \pi + {tan}^{ - 1}\bigg |\dfrac{y}{x}\bigg | \\ \\ \sf x - iy & \sf  - {tan}^{ - 1}\bigg |\dfrac{y}{x}\bigg | \end{array}} \\ \end{gathered}$