Write the expression in standard form. \[ \frac{7+i}{4-i} \]

To express \( \frac{7+i}{4-i} \) in standard form, we need to eliminate the imaginary unit \( i \) from the denominator. We do this by multiplying the numerator and denominator by the conjugate of the denominator, which is \( 4+i \). \[ \frac{7+i}{4-i} \cdot \frac{4+i}{4+i} = \frac{(7+i)(4+i)}{(4-i)(4+i)} \] Now, let's calculate the numerator: \[ (7+i)(4+i) = 7 \cdot 4 + 7 \cdot i + i \cdot 4 + i \cdot i = 28 + 7i + 4i + i^2 \] Since \( i^2 = -1 \): \[ 28 + 7i + 4i - 1 = 27 + 11i \] Next, we calculate the denominator: \[ (4-i)(4+i) = 4 \cdot 4 + 4 \cdot i - i \cdot 4 - i^2 = 16 + 0 + 1 = 17 \] Putting it all together, we have: \[ \frac{27 + 11i}{17} \] This can be separated into real and imaginary parts: \[ \frac{27}{17} + \frac{11}{17}i \] Thus, the expression in standard form is: \[ \frac{27}{17} + \frac{11}{17}i \]