Solving Division of Complex Numbers
The division problem shown in the image is asking you to divide a complex number, -5, by another complex number, -3 - i. To solve this, you can write the division as a fraction and then multiply the numerator and denominator by the conjugate of the denominator.
The conjugate of a complex number changes the sign of the imaginary part, so the conjugate of -3 - i is -3 + i.
Here's how you do it step by step:
\[\frac{-5}{-3 - i} \times \frac{-3 + i}{-3 + i} = \frac{-5(-3 + i)}{(-3 - i)(-3 + i)}\]
Now multiply the numerators and the denominators:
\[= \frac{15 - 5i}{(-3)^2 - (i)^2}\]
Since \(i^2 = -1\), the denominator becomes:
\[(-3)^2 - (-1) = 9 + 1 = 10\]
Thus the expression simplifies to:
\[= \frac{15 - 5i}{10}\]
Now divide both real and imaginary parts by 10:
\[= \frac{15}{10} - \frac{5i}{10}\]
Reducing these fractions we have:
\[= \frac{3}{2} - \frac{1}{2}i\]
So the answer to the division problem is:
\[1.5 - 0.5i\] or in fraction form \[\frac{3}{2} - \frac{1}{2}i\].
