Question

evaluate i^ 67 in complex number​

Answer 1

Answer:

Answer:

i^{37}+\frac{1}{i^{67}}=2ii

37

+

i

67

1

=2i

Step-by-step explanation:

Given : Expression i^{37}+\frac{1}{i^{67}}

To find : Evaluate the expression ?

Solution :

We can write,

i^{37}+\frac{1}{i^{67}}=i^{36}\cdot i+\frac{1}{i^{66}\cdot i}i

37

+

i

67

1

=i

36

⋅i+

i

66

⋅i

1

We know that, i=\sqrt{-1}\Rightarrow i^2=-1i=

−1

⇒i

2

=−1

So, i^{36}=i^{2\cdot 18}=(-1)^{18}=1 i

36

=i

2⋅18

=(−1)

18

=1

i^{66}=i^{2\cdot 33}=(-1)^{33}=-1i

66

=i

2⋅33

=(−1)

33

=−1

Substitute back,

=i+\frac{1}{-i}=i+

−i

1

=i-\frac{1}{i}=i−

i

1

=\frac{i^2-1}{i}=

i

i

2

−1

Multiply and divide by i,

=\frac{-1-1}{i}\times \frac{i}{i}=

i

−1−1

×

i

i

=\frac{-2i}{i^2}=

i

2

−2i

=2i=2i

Therefore, i^{37}+\frac{1}{i^{67}}=2ii

37

+

i

67

1

=2i