To simplify the expression $\frac{4i}{5 + i}$, we need to multiply both the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of $5 + i$ is $5 - i$. This method helps eliminate the imaginary part from the denominator.

So, we multiply both the numerator and denominator by $5 - i$:

$\frac{4i}{5 + i} \times \frac{5 - i}{5 - i} = \frac{4i(5 - i)}{(5 + i)(5 - i)}$

Now, simplify both the numerator and denominator:

Denominator:

The denominator is the product of a complex number and its conjugate: $(5 + i)(5 - i) = 5^2 - i^2 = 25 - (-1) = 25 + 1 = 26$

Numerator:

For the numerator, distribute $4i$ over $(5 - i)$: $4i(5 - i) = 4i \times 5 - 4i \times i = 20i - 4i^2$ Since $i^2 = -1$, we get: $20i - 4(-1) = 20i + 4$

So, the expression becomes: $\frac{20i + 4}{26}$

Finally, separate the real and imaginary parts: $\frac{20i}{26} + \frac{4}{26} = \frac{10i}{13} + \frac{2}{13}$

Thus, the simplified expression is: $\frac{2}{13} + \frac{10i}{13}$

Final Answer:

$\frac{2}{13} + \frac{10i}{13}$

Would you like further details on any of the steps or have any questions? Here are some related questions:

1. How do you multiply complex numbers?
2. What is a complex conjugate, and why do we use it?
3. How do you simplify complex fractions?
4. What are the properties of $i$ in complex numbers?
5. How do you handle dividing by a complex number in general?

Tip: When simplifying complex fractions, always use the complex conjugate of the denominator to remove the imaginary part.