Compute $|z|$ , $z = \frac{(2+i)^7(1-2i)^3}{(1+2i)^8}$

Question

Compute $|z|$ , $z = \frac{(2+i)^7(1-2i)^3}{(1+2i)^8}$,

if $z = a+ib$ then, I tried to do that with $|z| = (a+ib)(a-ib)$ then i multipled it $z$ with $z^-$ and then I got stuck. answer is $|z| = 5$

Answer 1

Hint. One may write $$ |z|^2=z\cdot \bar z=\frac{|2+i|^{14}|1-2i|^6}{|1+2i|^{16}}=\frac{(2^2+1^2)^7(1^2+2^2)^3}{(1^2+2^2)^8}. $$

Answer 2

hint: Use the following formulas: $||z_1z_2|| = ||z_1||||z_2||$, $||\frac{z_1}{z_2}|| = \dfrac{||z_1||}{||z_2||}$,and $||z^n|| = ||z||^n$. Can you continue?