Question

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

$\overline{z+w}$

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 add the given complex numbers and then find the complex conjugate of the resulting number as follows

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

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