What is the absolute value of: (1 + i) (6 + 2i)/ (4 + i) (9 + 3i)

I would simplify the fraction first, by multiplying the
numerator and denominator separately.  Don't forget to use FOIL when multiplying 2
complex numbers!!

(1+i)(6+2i) = 6+2i+6i+2i^2 =6 + 8i -2 = 4
+ 8i

(4+i)(9+3i) = 36+12i+9i+3i^2 = 36 + 21i -3 = 33
+21i

You can't reduce, there's no matching factors in
numerator & denominator.  I'm going to factor, just so I don't have to use large
numbers.  If you use a calculator, you may just use the numbers
there...

4(1+2i)/3(11+7i)

I
need to multiply the numerator and denominator by the conjugate of the denominator, so
it goes to a whole number.  The conjugate of (11+7i) is
(11-7i)

4(1+2i)(11-7i) = 4(11-7i+22i-14i^2) = 4(11+15i+14)
= 4(25+15i)

3(11+7i)(11-7i) = 3(121-49i^2) = 3(121+49) =
3(170)

I'm going to factor my numbers again, so I can
reduce the fraction:

4(5)(5+3i)/3(10)17    =
20(5+3i)/30(17)  = 2(5+3i)/3(17)

My simplified
fraction would be (10 + 6i)/51.  Now we can apply the absolute value of a complex
number:

sqrt[(100+36)/51] = sqrt (136/51).  It now depends
on what your teacher accepts as an answer.  The real answer is irrational, and if your
teacher accepts approximations to a certain decimal point, use your calculator and do
it.

If your teacher accepts nothing less than an exact
answer, you have to rationalize the denominator by multiplying numerator and denominator
by sqrt 51.  Then the exact answer is:

(sqrt
6936)/51.

Good luck!