Determine the modulus of the complex number z=(3-i)/(3-2i)

The presence of the complex number from denominator is not
allowed. According to the rule, we'll transform the complex number from denominator into
a real number. For this reason, we'll multiply the ratio by the conjugate of the
denominator.

The conjugate of denominator is
3+2i.

The ratio will
become:

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

We'll notice that the denominator
is a difference of squares:

(3-2i)(3+2i) = 9  -
4i^2

But i^2 = -1

(3-2i)(3+2i)
=  9 + 4

(3-2i)(3+2i) =
13

We'll remove the brackets from
numerator:

(3-i)(3+2i) = 9 + 6i - 3i -
2i^2

We'll combine like
terms:

(3-i)(3+2i) = 9 + 3i +
2

(3-i)(3+2i) = 11 + 3i

The
complex number z is:

z = (11 +
3i)/13

z = 11/13 + 3i/13

The
modulus of the complex number is:

|z| = sqrt [Re(z)^2 +
Im(z)^2]

|z| = sqrt[(11/13)^2 +
(3/13)^2]

|z| = sqrt
(121+9)/13

|z| = sqrt
(130)/13