Question

Express 1/i+2/i2+3/i3+5/i4 in the from a/ib​

Answer 1

Question →

$\frac{1}{i} + \frac{2}{ {i}^{2} } + \frac{3}{ {i}^{3} } + \frac{4}{ {i}^{4} }$

Solution →

As we know that

i² = -1

i³ = -i

i⁴ = 1

So putting these values

$= > \frac{1}{i} + ( \frac{2}{ - 1} ) + (\frac{3}{ - i} ) + \frac{5}{1}$

$= > (\frac{1}{i} - \frac{3}{i} ) + 5 - 2$

$= > \frac{1 - 3}{i} + 3$

$= > \frac{2}{i} + 3$

Now taking ( i ) as LCM

$= > \frac{2 + 3i}{i}$

Now Multiplying and dividing by ( -i)

$= > \frac{2 + 3i}{i} \times \frac{ - i}{ - i}$

$= > \frac{ - 2i - {3i}^{2} }{ - {i}^{2} }$

Again putting the value of i² here

$= > \frac{ - 2i - (3 \times - 1)}{ - ( - 1) }$

$= > \frac{3 - 2i}{1} = 3 - 2i$

Hence , a+ib = 3 - 2i

Hope it helps

Answer 2

Answer:

i² = -1 , i³ = (-i) & i⁴ = 1

using all we get ,

1/i -2 -3/i +5

(1/i -3/i) +3

2/i +3

(2+3i)/i

Rationalizing it we get,

(2+3i)(-i)/(-i)²

(-2i-3i²)

(3-2i) (Ans)