Question

If $z=\frac{(\sqrt{3}+i{)}^3(3i+4{)}^2}{(8+6i{)}^2}$, then $|z|$ is equal to

• A. $8$
• B. $2$
• C. $5$
• D. $4$
• E. $10$

Answer 1

The correct option is B $2$

Given, $z=\frac{(\sqrt{3}+i{)}^3(3i+4{)}^2}{(8+6i{)}^2}$

$=\frac{(3\sqrt{3}+i^3+9i+3\sqrt{3}{i}^2)(9i^2+16+24i)}{64+36i^2+96i}$

$=\frac{(3\sqrt{3}-i+9i-3\sqrt{3})(-9+16+24i)}{64-36+96i}$

$=\frac{8i(7+24i)}{28+96i}$

$=\frac{8i(7+24i)}{4(7+24i)}$

$=2i$

Thus $|z|=|2i|=2$