Question

if ( w = 5 - 8i ), write the expression in the standard form ( a + bi ).( w + overline{w} )( w + overline{w} = square ) (simplify your answer.)

Explanation:

Step1: Find the conjugate of \( w \)

Given \( w = 5 - 8i \), the conjugate of a complex number \( a + bi \) is \( a - bi \) (or vice - versa). So the conjugate of \( w \), denoted as \( \overline{w} \), is \( 5+8i \).

Step2: Add \( w \) and \( \overline{w} \)

We need to calculate \( w+\overline{w}=(5 - 8i)+(5 + 8i) \).
Combine the real parts and the imaginary parts separately. The real parts are \( 5\) and \( 5\), and the imaginary parts are \( - 8i\) and \( 8i\).
For the real parts: \( 5 + 5=10 \).
For the imaginary parts: \( - 8i+8i = 0 \).
So \( w+\overline{w}=10+0i = 10 \).

Answer:

\( 10 \)