First, let's simplify the expression step by step.
Expanding the first part: (1+i)(1-2i= 1(1) + 1(-2i) + i(1) + i(-2i= 1 - 2i + i - 2i^= 1 - 2i + i + = 3 - i
Expanding the second part: (1+i)(1+2i= 1(1) + 1(2i) + i(1) + i(2i= 1 + 2i + i + 2i^= 1 + 2i + i - = -1 + 3i
Now, we substitute these results back into the original expression:
(3 - i + (-1 + 3i)) / (1+2i)^2 - (1-2i)
= (2 + 2i) / (1+2i)^2 - (1-2i)
To simplify further, we need to square the denominator (1+2i)^2:
(1+2i)^2 = (1+2i)(1+2i= 11 + 12i + 2i1 + 2i2= 1 + 2i + 2i + 4i^= 1 + 4i + 4i - = -3 + 8i
Now substitute back:
(2 + 2i) / (-3 + 8i) - (1-2i)
Now, let's simplify this expression further if needed.
First, let's simplify the expression step by step.
Expanding the first part: (1+i)(1-2i
= 1(1) + 1(-2i) + i(1) + i(-2i
= 1 - 2i + i - 2i^
= 1 - 2i + i +
= 3 - i
Expanding the second part: (1+i)(1+2i
= 1(1) + 1(2i) + i(1) + i(2i
= 1 + 2i + i + 2i^
= 1 + 2i + i -
= -1 + 3i
Now, we substitute these results back into the original expression:
(3 - i + (-1 + 3i)) / (1+2i)^2 - (1-2i)
= (2 + 2i) / (1+2i)^2 - (1-2i)
To simplify further, we need to square the denominator (1+2i)^2:
(1+2i)^2 = (1+2i)(1+2i
= 11 + 12i + 2i1 + 2i2
= 1 + 2i + 2i + 4i^
= 1 + 4i + 4i -
= -3 + 8i
Now substitute back:
(2 + 2i) / (-3 + 8i) - (1-2i)
Now, let's simplify this expression further if needed.