To find the Modulus of a complex number

Question

I was given the following expression ,

$|\frac{(3+i)(2-i)}{(1+i)}|$ ,

and was asked to find its value

This is how I proceeded ,

On solving the numerator , the given expression transforms to $|\frac{7-i}{1+i}|$ Then I took the conjugate of the denominator and finally got the expression

$\frac{8-8i}{2}$

$= 4|(1-i)|$

Now according to me it’s modulus should be $4\sqrt{2}$ however the correct answer is $5$ . Could you please correct me where I am mistaken ? And please suggest a method to solve this. Thank you .

Answer 1

Because $$\left|\frac{(3+i)(2-i)}{(1+i)}\right|=\frac{\sqrt{10}\cdot\sqrt{5}}{\sqrt2}=5.$$

Answer 2

note that $$\frac{(3+i)(2-i)}{1+i}=3-4i$$

Answer 3

$(7-i)(1-i) \ne 8-8i$ but $(7-i)(1-i) = 6-8i$. So $\big|\frac{6-8i}{2} \big| = 5$.

Easier way to evaluate is seperating the absolute value as: $$\bigg|\frac{(3+i)(2-i)}{(1+i)}\bigg| = \frac{|3+i| |2-i|}{|1+i|} = \frac{\sqrt{10} \cdot \sqrt{5}}{\sqrt{2}} = 5$$