Question

If $z=4-3\sqrt{2}i$ then the value of $z\overline{z}$

Answer 1

This is a question of a complex number and we need to use the concepts of conjugate of complex numbers. Furthermore, we need to be in good touch with the multiplication of complex numbers.

Complete step by step answer:
A complex number consists of an imaginary part and a real part. The part with the $i$ is the imaginary part. The given complex number is $z=4-3\sqrt{2}i$and here, $4$ is the real part and $3\sqrt{2}$ is the imaginary part. And we are to find out the multiplication of the complex number with its conjugate.
But first of all, what is the conjugate of a complex number? It is the same number with the opposite sign of the imaginary part. This means, if we have a complex number of the form $a+ib$ then its conjugate will be $a-ib$. The conjugate of a complex number $z$ is denoted with $\overline{z}$. So the conjugate of $z=4-3\sqrt{2}i$ is $4+3\sqrt{2}i$.
Now, we need to multiply the two complex numbers. The multiplication of complex numbers is the same as normal numbers but we need to keep in mind the square of $i$ or multiplication of $i$ in general. So the required form is $\left( 4-3\sqrt{2}i \right)\left( 4+3\sqrt{2}i \right)$
And this becomes $(4\times 4)+4\times 3\sqrt{2}i+4\times \left( -3\sqrt{2}i \right)+\left( -3\sqrt{2}i \right)3\sqrt{2}i$
Now the two terms in the middle get cancelled and Therefore the final answer after solving the term is $16+18$=$34$.

Note: i is $\sqrt{-1}$ and when we multiply $i$ with $i$, we multiply $\sqrt{-1}$with $\sqrt{-1}$ and that is $-1$. Which is why the final answer to the question is without any $i$. So it is very important to take care of $i$ while multiplying.