calculate |z| when 2+6i=4z+8z'

The expression of the module of a complex
number:

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

For the complex number z, written
algebraically:

z = a + b*i, we'll
have

the real part = Re(z) =
a

and

the imaginary part is
Im(z) = b.

|z| = sqrt(a^2 +
b^2)

The complex number z' is the conjugate of z and it's
expression is z' = a - b*i

Now, we'll re-write the given
expression:

2+6i=4z+8z'

We'll
factorize by 2, both sides:

2(1+3i) =
2(2z+4z')

We'll divide by 2 both
sides:

1+3i = 2z+4z'

We'll
substitute z and z' into the expression above:

1+3i =
2a+2b*i + 4a - 4b*i

We'll combine like terms from the right
side:

1+3i = 6a - 2b*i

The
real part from the left side has to be equal to the real part from the right
side.

6a = 1

We'll divide by
6:

a =
1/6

The imaginary part from the left side
has to be equal to the imaginary part from the right
side.

3*i=
-2b*i

3=-2b

We'll divide by
-2:

b =
-3/2

The module of the complex number z
is:

|z| = sqrt (1/36 +
9/4)

|z| = sqrt
(82/36)

|z| = [sqrt (82)] /
6