Question

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

Answer 1

Дробь (3+2i)*(2-i)/2+3i *(1+i)

Дробь 6-3i+4i+2+8i+i^2/2+3i

Дробь 6-3i+4i+2+8i-1/2+3i

7+9i/2+3i

41/13-3/13i

Answer 2

To simplify the expression, let's perform the multiplication and division step by step:

First, let's multiply the numerators and denominators separately:

Numerator: (3 + 2i) * (2 - i) = 3 * 2 + 3 * (-i) + 2i * 2 + 2i * (-i) = 6 - 3i + 4i - 2i^2

Since i^2 = -1: = 6 + i + 4i + 2 = 8 + 5i

Denominator: (2 + 3i) * (1 + i) = 2 * 1 + 2 * i + 3i * 1 + 3i * i = 2 + 2i + 3i + 3i^2

Again, since i^2 = -1: = 2 + 2i + 3i - 3 = -1 + 5i

Now, let's divide the numerator by the denominator:

(8 + 5i) / (-1 + 5i)

To divide complex numbers, we multiply the numerator and denominator by the conjugate of the denominator.

The conjugate of (-1 + 5i) is (-1 - 5i).

Multiplying the numerator and denominator by the conjugate:

(8 + 5i) * (-1 - 5i) / (-1 + 5i) * (-1 - 5i)

Expanding the numerator and denominator:

= (-8 - 40i - 5i + 25i^2) / (1 + 5i - 5i - 25i^2)

Since i^2 = -1:

= (-8 - 40i - 5i + 25 * (-1)) / (1 + 5i - 5i - 25 * (-1))

Simplifying:

= (-8 - 40i - 5i - 25) / (1 + 5i - 5i + 25)

= (-33 - 45i) / 26

= -33/26 - 45i/26

Therefore, the simplified form of the expression is approximately -1.269 + (-1.731i).