# How do you simplify (3i)/(9-6i)?

Dec 31, 2015

$- \frac{2}{13} + \frac{3}{13} i$

#### Explanation:

Divide each term by $3$.

$\implies \frac{i}{3 - 2 i}$

Now, to remove the imaginary part from the denominator and write the answer as a complex number in the form $a + b i$, multiply the fraction by the complex conjugate of the denominator.

$\implies \frac{i}{3 - 2 i} \left(\frac{3 + 2 i}{3 + 2 i}\right)$

Distribute. Notice that the bottom will form a difference of squares.

$\implies \frac{3 i + 2 {i}^{2}}{9 - 4 {i}^{2}}$

To continue simplifying, rewrite ${i}^{2}$ as $- 1$. ${i}^{2} = - 1$ since $i = \sqrt{- 1}$.

$\implies \frac{3 i + 2 \left(- 1\right)}{9 - 4 \left(- 1\right)} = \frac{- 2 + 3 i}{13}$

Split apart the numerator to write in $a + b i$ form.

$\implies - \frac{2}{13} + \frac{3}{13} i$