Question

divide and express the result in standard form.
\\(\frac{9}{6 - i}\\)
\\(\frac{9}{6 - i} = \square\\) (simplify your answer. type your answer in the form \\(a + bi\\).)

Explanation:

Step1: Multiply numerator and denominator by conjugate

To divide complex numbers, we multiply the numerator and denominator by the conjugate of the denominator. The conjugate of \(6 - i\) is \(6 + i\). So we have:

$$ \frac{9}{6 - i} \times \frac{6 + i}{6 + i} $$

Step2: Multiply numerators and denominators

First, multiply the numerators: \(9\times(6 + i)=54 + 9i\).
Then, multiply the denominators using the formula \((a - b)(a + b)=a^2 - b^2\), where \(a = 6\) and \(b = i\). So \((6 - i)(6 + i)=6^2 - i^2=36 - (-1)=36 + 1 = 37\).
Now our expression is \(\frac{54 + 9i}{37}\).

Step3: Split into real and imaginary parts

We can split this fraction into two parts: \(\frac{54}{37}+\frac{9}{37}i\).

Answer:

\(\frac{54}{37}+\frac{9}{37}i\)