Question

divide and express the result in standard form.
\\(37)\dfrac{8}{6 + i}\\)

Explanation:

Step1: Rationalize the denominator

To divide \(\frac{8}{6 + i}\), we multiply the numerator and denominator by the conjugate of the denominator, which is \(6 - i\).

$$ \frac{8}{6 + i} \times \frac{6 - i}{6 - i} $$

Step2: Multiply the numerators and denominators

First, multiply the numerators: \(8\times(6 - i)=48 - 8i\)
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 we have \(\frac{48 - 8i}{37}\)

Step3: Split the fraction

We can split this fraction into two parts: \(\frac{48}{37}-\frac{8}{37}i\)

Answer:

\(\frac{48}{37}-\frac{8}{37}i\)