Question

Simplify $\Big[\frac{1}{1-2i}+\frac{3}{1+i}\Big] \Big[\frac{3+4i}{2-4i}\Big]$.

Answer 1

Answer:

( 1 + 9i)/4

Step-by-step explanation:

Hi,

Consider 1/1-2i

Multiplying and dividing by 1 + 2i, we get

(1 + 2i)/((1 + 2i)(1 - 2i)

= (1 + 2i)/5-------------------------------------------(1)

Consider 3/1 + i

Multiplying and dividing by 1 - i, we get

3(1 - i)/(1 + i)*(1 - i)

= 3(1 - i)/2

= 3/2 -3i/2------------------------------------------(2)

Consider 1/1-2i + 3/1 + i

From (1) and (2), we get

= (1 + 2i)/5 + 3/2 -3i/2

= (1/5 + 3/2) + i(2/5 - 3/2)

= 17/10 + i(-11/10)

= 17/10 - 11i/10--------------------------------------(3)

Consider (3 + 4i)/(2 - 4i)

Multiplying  and dividing by 2 + 4i, we get

[(3 + 4i)*(2 + 4i)/(2 - 4i)*2 + 4i]

= ( 6 - 16 + 12i + 8i)/(4 + 16)

= ( -10 + 20i)/20

= (-1 + 2i)/2---------------------------------------------(4)

Consider [1/1-2i + 3/1 + i]*[(3 + 4i)/(2 - 4i)]

From (3) and (4), we get

(17/10 - 11i/10)*( (-1 + 2i)/2)

= (17 - 11i)*(-1 + 2i)/20

= ( -17 + 22 + 34i + 11i)/20

= (5 + 45i)/20

= ( 1 + 9i)/4

Hope, it helps !