Write in Standard Form (6i)/(3-i)

$\frac{6 i}{3 - i}$

Step 1

Multiply the numerator and denominator of $\frac{6 i}{3 - i}$ by the conjugate of $3 - i$ to make the denominator real.

$\frac{6 i}{3 - i} \cdot \frac{3 + i}{3 + i}$

Step 2

Multiply.

Combine.

$\frac{6 i (3 + i)}{(3 - i) (3 + i)}$

Simplify the numerator.

Apply the distributive property.

$\frac{6 i \cdot 3 + 6 i i}{(3 - i) (3 + i)}$

Multiply $3$ by $6$ .

$\frac{18 i + 6 i i}{(3 - i) (3 + i)}$

Multiply $6 i i$ .

Raise $i$ to the power of $1$ .

$\frac{18 i + 6 (i^{1} i)}{(3 - i) (3 + i)}$

Raise $i$ to the power of $1$ .

$\frac{18 i + 6 (i^{1} i^{1})}{(3 - i) (3 + i)}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{18 i + 6 i^{1 + 1}}{(3 - i) (3 + i)}$

Add $1$ and $1$ .

$\frac{18 i + 6 i^{2}}{(3 - i) (3 + i)}$

Simplify each term.

Rewrite $i^{2}$ as $- 1$ .

$\frac{18 i + 6 \cdot - 1}{(3 - i) (3 + i)}$

Multiply $6$ by $- 1$ .

$\frac{18 i - 6}{(3 - i) (3 + i)}$

Reorder $18 i$ and $- 6$ .

$\frac{- 6 + 18 i}{(3 - i) (3 + i)}$

Simplify the denominator.

Expand $(3 - i) (3 + i)$ using the FOIL Method.

Apply the distributive property.

$\frac{- 6 + 18 i}{3 (3 + i) - i (3 + i)}$

Apply the distributive property.

$\frac{- 6 + 18 i}{3 \cdot 3 + 3 i - i (3 + i)}$

Apply the distributive property.

$\frac{- 6 + 18 i}{3 \cdot 3 + 3 i - i \cdot 3 - i i}$

Simplify.

Multiply $3$ by $3$ .

$\frac{- 6 + 18 i}{9 + 3 i - i \cdot 3 - i i}$

Multiply $3$ by $- 1$ .

$\frac{- 6 + 18 i}{9 + 3 i - 3 i - i i}$

Raise $i$ to the power of $1$ .

$\frac{- 6 + 18 i}{9 + 3 i - 3 i - (i^{1} i)}$

Raise $i$ to the power of $1$ .

$\frac{- 6 + 18 i}{9 + 3 i - 3 i - (i^{1} i^{1})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{- 6 + 18 i}{9 + 3 i - 3 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{- 6 + 18 i}{9 + 3 i - 3 i - i^{2}}$

Subtract $3 i$ from $3 i$ .

$\frac{- 6 + 18 i}{9 + 0 - i^{2}}$

Add $9$ and $0$ .

$\frac{- 6 + 18 i}{9 - i^{2}}$

Simplify each term.

Rewrite $i^{2}$ as $- 1$ .

$\frac{- 6 + 18 i}{9 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{- 6 + 18 i}{9 + 1}$

Add $9$ and $1$ .

$\frac{- 6 + 18 i}{10}$

Step 3

Cancel the common factor of $- 6 + 18 i$ and $10$ .

Factor $2$ out of $- 6$ .

$\frac{2 (- 3) + 18 i}{10}$

Factor $2$ out of $18 i$ .

$\frac{2 (- 3) + 2 (9 i)}{10}$

Factor $2$ out of $2 (- 3) + 2 (9 i)$ .

$\frac{2 (- 3 + 9 i)}{10}$

Cancel the common factors.

Factor $2$ out of $10$ .

$\frac{2 (- 3 + 9 i)}{2 \cdot 5}$

Cancel the common factor.

$\frac{2 (- 3 + 9 i)}{2 \cdot 5}$

Rewrite the expression.

$\frac{- 3 + 9 i}{5}$

Step 4

Split the fraction $\frac{- 3 + 9 i}{5}$ into two fractions.

$\frac{- 3}{5} + \frac{9 i}{5}$

Step 5

Move the negative in front of the fraction.

$- \frac{3}{5} + \frac{9 i}{5}$

Write in Standard Form (6i)/(3-i)

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