# How do you write the following expression in standard form (1+i)/i-3/(4-i)?

Oct 22, 2016

The standard form is $\frac{5}{17} - \frac{20 i}{17}$

#### Explanation:

le $z = \frac{1 + i}{i} - \frac{3}{4 - i}$

Reducing to the same denominator

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

Recall that ${i}^{2} = - 1$

Then $z = \frac{4 + 3 i - {i}^{2} - 3 i}{4 i - {i}^{2}} = \frac{5}{1 + 4 i}$

We simplify further by multiplying by the conjugate of the denominator $1 - 4 i$

$z = \frac{5 \cdot \left(1 - 4 i\right)}{\left(1 + 4 i\right) \left(1 - 4 i\right)} = \frac{5 - 20 i}{1 - 16 {i}^{2}} = \frac{5 - 20 i}{17}$

$z = \frac{5}{17} - \frac{20 i}{17}$