Question

If $z=2+\sqrt{3}i$, then find the value of $z.\overline{z}$.

Answer 1

Evaluate the required expression:

The given expression is $z=2+\sqrt{3}i$.

We know that for a complex number, say $a=m+in$ the conjugate is

$\overline{a}=m-in$

$\therefore \overline{z}=2-\sqrt{3}i$

The required value is

$z.\overline{z}=\left(2+\sqrt{3}i\right)\left(2-\sqrt{3}i\right)$

$=4-3i^2$ $\left[\because i^2=-1;\left(a+b\right)\left(a-b\right)=a^2-b^2\right]$

$=4+3$

$=7$

Hence, the value of $z.\overline{z}$ is $7$.