What is the simplified form of the expression (3 + i)/(4 - i)?

To simplify the expression ((3 + i)/(4 - i)), we start by multiplying both the numerator and the denominator by the conjugate of the denominator, which is ((4 + i)). This technique helps eliminate the imaginary part from the denominator.

Here’s how the multiplication looks:

[

\frac{3 + i}{4 - i} \cdot \frac{4 + i}{4 + i} = \frac{(3 + i)(4 + i)}{(4 - i)(4 + i)}

]

First, let's simplify the denominator:

[

(4 - i)(4 + i) = 4^2 - i^2 = 16 - (-1) = 16 + 1 = 17

]

Now, let's expand the numerator:

[

(3 + i)(4 + i) = 3 \cdot 4 + 3 \cdot i + i \cdot 4 + i \cdot i = 12 + 3i + 4i + i^2 = 12 + 7i - 1 = 11 + 7i

]

Putting it all together, we have:

[

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