Question

If $z_1=2\sqrt2(1+i)$ and $z_2=1+i\sqrt3$ , then $z_1^2\,\,z_2^3$ is equal to

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

Answer 1

Answer: (D)

$-128i$

Answer 2

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