Question

i³⁸² + i³⁸⁰ + i³⁷⁸+ i³⁷⁴ ÷ i³⁷² + i³⁷⁰ +i³⁶⁸ +i³⁶⁴ is

Answer 1

${i}^{2} = - 1$

$</p><p>{i}^{4} = {i}^{2} \times {i}^{2} = 1$

${i}^{364} = {( {i}^{4} )}^{91} = {1}^{91} = 1$

Also,

${i}^{368} = {i}^{364} \times {i}^{4} = 1 \times 1 = 1$

Similarly

${i}^{370} = {i}^{368} \times {i}^{2} = 1 \times - 1 = - 1$

${i}^{372} = {i}^{368} \times {i}^{4} = 1 \times 1 = 1$

${i}^{374} = - 1$

${i}^{378} = - 1$

${i}^{380} = 1$

${i}^{382} = - 1$

Hence given expression is

$\frac{ - 2}{2} = - 1$

Answer 2

Answer:

=−1

{i}^{4} = {i}^{2} \times {i}^{2} = 1i

4

=i

2

×i

2

=1

{i}^{364} = {( {i}^{4} )}^{91} = {1}^{91} = 1i

364

=(i

4

)

91

=1

91

=1

Also,

{i}^{368} = {i}^{364} \times {i}^{4} = 1 \times 1 = 1i

368

=i

364

×i

4

=1×1=1

Similarly

{i}^{370} = {i}^{368} \times {i}^{2} = 1 \times - 1 = - 1i

370

=i

368

×i

2

=1×−1=−1

{i}^{372} = {i}^{368} \times {i}^{4} = 1 \times 1 = 1i

372

=i

368

×i

4

=1×1=1

{i}^{374} = - 1i

374

=−1

{i}^{378} = - 1i

378

=−1

{i}^{380} = 1i

380

=1

{i}^{382} = - 1i

382

=−1