Question

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

$\overline{z}\cdot \overline{w}$

Short answer

The value of the given expression is $23+14i$.

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 multiplication of complex numbers.

To multiply two complex numbers, multiply them as binomial expressions and use the result of imaginary number, $i^2=-1$.

The FOIL method of multiplying binomials is

$\left(a+b\right)+\left(c+d\right)=a\cdot c+a\cdot d+b\cdot c+b\cdot d$

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

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

Step 3. Evaluate the expression.

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

$$\begin{gathered} \overline{z}\cdot \overline{w}=\left(\overline{3-4i}\right)\cdot \left(\overline{5+2i}\right) \\ =\left(3+4i\right)\cdot \left(5-2i\right) \\ =3\cdot 5+3\cdot \left(-2i\right)+4i\cdot \left(5\right)+4i\cdot \left(-2i\right) \\ =15-6i+20i-8i^2 \\ =15+14i-8\left(-1\right) \\ =15+14i+8 \\ =23+14i \end{gathered}$$

Thus, the value of the expression is $23+14i$.