Calculate the absolute value of $z=(10+5i)(1+10i)(4+2i)(5+2i)$

Question

I am doing some repetition of complex numbers and I got to this question:

Calculate the absolute value of $z=(10+5i)(1+10i)(4+2i)(5+2i)$

My approach has been to first multiply the imaginary numbers and then I get: $z = -2530 + 960i$

I am not supposed to use a calculator so this is not an easy number to calculate the absolute value of. There must be an easier (smarter) way of doing this?

Thank you!

Answer 1

The absolute value of a product is the product of the absolute values, hence

$$|z|=|(10+5i)(1+10i)(4+2i)(5+2i)|=|10+5i|\cdot |1+10i|\cdot|4+2i|\cdot|5+2i|\\ =\sqrt{125}\cdot\sqrt{101}\cdot\sqrt{20}\cdot\sqrt{29}=\sqrt{5^3}\cdot\sqrt{101}\cdot\sqrt{2^2\cdot5}\cdot\sqrt{29}=50\sqrt{29\cdot101}=50\sqrt{2929}$$

Answer 2

We can prove $$\left|\prod_{i=1}^n a_i\right|=\prod_{i=1}^n|a_i|$$

as $$(a+ib)(c+id)=(ac-bd)+i(ad+bc)$$

and $$(a^2+b^2)(c^2+d^2)=(ac-bd)^2+(ad+bc)^2$$

See also : Brahmagupta–Fibonacci identity

Answer 3

You can notice that $10+5i=5(2+i)$ and $4+2i=2(2+i)$ and use that absolute value is multiplicative to get that the magnitude of the product of these is $50$, then multiply by the magnitudes is the others.