# How do you simplify (3+i)/(1+4i)?

Dec 3, 2016

There are several ways to do it. I like to multiply numerator and denominator by the complex conjugate of the denominator.

#### Explanation:

Multiply numerator and denominator by the complex conjugate of the denominator:

$\frac{3 + i}{1 + 4 i} \frac{1 - 4 i}{1 - 4 i}$

Use the pattern, $\left(a + b\right) \left(a - b\right) = {a}^{2} - {b}^{2}$, to multiply the denominator:

$\frac{\left(3 + i\right) \left(1 - 4 i\right)}{1 - 16 {i}^{2}}$

Use ${i}^{2} = - 1$ to make the denominator a real number:

$\frac{\left(3 + i\right) \left(1 - 4 i\right)}{17}$

Use the F.O.I.L. method to multiply the numerator.

$\frac{3 - 12 i + i - 4 {i}^{2}}{17}$

Use ${i}^{2} = - 1$:

$\frac{3 - 12 i + i + 4}{17}$

Combine like terms:

$\frac{7 - 11 i}{17}$

Distribute the 17:

$\frac{7}{17} - \frac{11}{17} i$