Question

Evaluate the given expression for $z=3-4i\text{and}w=5+2i$.

$\overline{z}+\overline{w}$

Short answer

The value of the given expression is $8+2i$.

Step-by-step solution

Step 1. Concept of complex conjugate.

The complex conjugate of the complex number$a+bi$ is represented by a horizontal bar over it, that is, the complex conjugate of$a+bi$ is denoted by $\overline{a+bi}$.

The complex conjugate of any complex number can be obtained by changing the sign of the imaginary part of the complex number. The complex conjugate of $a+bi$is $a-bi$.

The product of a complex number and its complex conjugate is given by

$\left(a+bi\right)\left(a-bi\right)=a^2+b^2$

The imaginary number i is defined as the square root of negative one. That is, $i^2=-1$.

Step 2. Concept of addition of complex numbers.

To add or subtract two complex numbers, simply add/subtract the real parts together and the imaginary parts together. That is,

$\left(a+bi\right)+\left(c+di\right)=\left(a+c\right)+\left(b+d\right)i$

Step 3. Evaluate the expression.

To find the value of the given expression, first find the complex conjugates of the given complex numbers and then add them as shown below

$$\begin{gathered} \overline{z}+\overline{w}=\left(\overline{3-4i}\right)+\left(\overline{5+2i}\right) \\ =\left(3+4i\right)+\left(5-2i\right) \\ =\left(3+5\right)+\left(4+\left(-2\right)\right)i \\ =\left(8\right)+\left(2\right)i \\ =8+2i \end{gathered}$$

Thus, the value of the expression is $8+2i$.