Question

If $z_1=2\sqrt{2}(1+i)$ and $z_2=1+i\sqrt{3}$ , then $z_1^2{z}_2^3$ is equal to

• A. $128i$
• B. $64i$
• C. $-64i$
• D. $-128i$

Answer 1

Answer: (D)

We have $z_1=2\sqrt{2}\left(1+i\right)$
$z_1^2=8{\left(1+i\right)}^{2}8\left(1+i^2+2i\right)=16i$
Also, $z_2=1+i\sqrt{3}$
$\Rightarrow z_2^3={\left(1+i\sqrt{3}i\right)}^3=1+3\sqrt{3}{i}^3+3\sqrt{3}i\left(1+i\sqrt{3}\right)$
$=1-3\sqrt{3}i+3\sqrt{3}i+9^2=-8$
So, $z_1^2{z}_2^3=16\left(-8\right)i=-128i$