Question:
If $z_1=2\sqrt2(1+i)$ and $z_2=1+i\sqrt3$ , then $z_1^2\,\,z_2^3$ is equal to
If $z_1=2\sqrt2(1+i)$ and $z_2=1+i\sqrt3$ , then $z_1^2\,\,z_2^3$ is equal to
Updated On: Jul 6, 2022
- $128i$
- $64i$
- $-64i$
- $-128i$
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The Correct Option is D
Solution and Explanation
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, $z^{2}_{1}z^{3}_{2}=16 \left(-8\right)i=-128i$
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Concepts Used:
Complex Numbers and Quadratic Equations
Complex Number: Any number that is formed as a+ib is called a complex number. For example: 9+3i,7+8i are complex numbers. Here i = -1. With this we can say that i² = 1. So, for every equation which does not have a real solution we can use i = -1.
Quadratic equation: A polynomial that has two roots or is of the degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b and c are the real numbers.
