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Q.
If $Z = \frac {(\sqrt {3}+ i)^3 (3i+4)^2}{{(8+6i)^2}}$ then $|Z|$ is equal to
KCETKCET 2015Complex Numbers and Quadratic Equations
Solution:
We have ,
$z=\frac{\left(\sqrt{3}+i\right)^{3}\left(3i+4\right)^{2}}{\left(8+6i\right)^{2}}$
$\Rightarrow \left|z\right|=\frac{\left|\sqrt{3}+i\right|^{3} \left|3\,i+4\right|^{2}}{\left|8+6\,i\right|^{2}}$
$=\frac{\left(\sqrt{\left(\sqrt{3}\right)^{2}+\left(1\right)^{2}}\right)^{3}\left(\sqrt{\left(3\right)^{2}+\left(4\right)^{2}}\right)^{2}}{\left(\sqrt{8^{2}+6^{2}}\right)^{2}}$
$=\frac{\left(2\right)^{3}\left(5\right)^{2}}{\left(10\right)^{2}}=2$