Question

let $z = 5 - 9i$ and $w = 6 + i$.

which expression shows the first step in finding the quotient of $\frac{z}{w}$?

$\bigcirc$ $\frac{5 - 9i}{6 + i} \cdot \frac{6 - i}{6 - i}$

$\bigcirc$ $\frac{5 - 9i}{6 + i} \cdot \frac{6 + i}{6 + i}$

$\bigcirc$ $\frac{5 - 9i}{6 + i} \cdot \frac{5 - 9i}{5 - 9i}$

$\bigcirc$ $\frac{5 - 9i}{6 + i} \cdot \frac{5 + 9i}{5 + 9i}$

Explanation:

Step1: Recall complex division rule

To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator. The conjugate of \(a+bi\) is \(a-bi\).

Step2: Identify denominator conjugate

For \(w = 6+i\), its conjugate is \(6-i\).

Step3: Apply to the quotient

The quotient \(\frac{z}{w} = \frac{5-9i}{6+i}\), so multiply by \(\frac{6-i}{6-i}\) (which equals 1, so value is unchanged).
<Expression>$\frac{5-9i}{6+i} \cdot \frac{6-i}{6-i}$</Expression>

Answer:

$\frac{5 - 9i}{6 + i} \cdot \frac{6 - i}{6 - i}$