The problem provided is to divide $\frac{6}{5 - i}$ and express the result in standard form (which is the form $a + bi$, where $a$ and $b$ are real numbers).

Step-by-Step Solution:

1. To divide by a complex number, multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of $5 - i$ is $5 + i$.

2. Multiply the numerator and the denominator by $5 + i$:

$\frac{6}{5 - i} \times \frac{5 + i}{5 + i} = \frac{6(5 + i)}{(5 - i)(5 + i)}$

3. Simplify the denominator using the difference of squares formula: $(5 - i)(5 + i) = 5^2 - i^2 = 25 - (-1) = 25 + 1 = 26$

4. Now, multiply the numerator: $6(5 + i) = 30 + 6i$

5. Therefore, the expression becomes: $\frac{30 + 6i}{26} = \frac{30}{26} + \frac{6i}{26} = \frac{15}{13} + \frac{3i}{13}$

Final Answer:

The result in standard form is: $\frac{15}{13} + \frac{3i}{13}$

Would you like further details or have any questions about the process? Here are some related questions to consider:

1. What is the general process for dividing complex numbers?
2. How do you find the conjugate of a complex number?
3. Why do we multiply by the conjugate when dividing complex numbers?
4. Can you generalize this division method to any two complex numbers?
5. How would the process change if the denominator had both real and imaginary components?

Tip: When working with complex numbers, always simplify the denominator using the difference of squares when multiplying by the conjugate.