Write in Standard Form (6+2i)/(9-3i)

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

Step 1

Multiply the numerator and denominator of $\frac{6 + 2 i}{9 - 3 i}$ by the conjugate of $9 - 3 i$ to make the denominator real.

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

Step 2

Multiply.

Combine.

$\frac{(6 + 2 i) (9 + 3 i)}{(9 - 3 i) (9 + 3 i)}$

Simplify the numerator.

Expand $(6 + 2 i) (9 + 3 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{6 (9 + 3 i) + 2 i (9 + 3 i)}{(9 - 3 i) (9 + 3 i)}$

Apply the distributive property.

$\frac{6 \cdot 9 + 6 (3 i) + 2 i (9 + 3 i)}{(9 - 3 i) (9 + 3 i)}$

Apply the distributive property.

$\frac{6 \cdot 9 + 6 (3 i) + 2 i \cdot 9 + 2 i (3 i)}{(9 - 3 i) (9 + 3 i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $6$ by $9$ .

$\frac{54 + 6 (3 i) + 2 i \cdot 9 + 2 i (3 i)}{(9 - 3 i) (9 + 3 i)}$

Multiply $3$ by $6$ .

$\frac{54 + 18 i + 2 i \cdot 9 + 2 i (3 i)}{(9 - 3 i) (9 + 3 i)}$

Multiply $9$ by $2$ .

$\frac{54 + 18 i + 18 i + 2 i (3 i)}{(9 - 3 i) (9 + 3 i)}$

Multiply $2 i (3 i)$ .

Multiply $3$ by $2$ .

$\frac{54 + 18 i + 18 i + 6 i i}{(9 - 3 i) (9 + 3 i)}$

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

$\frac{54 + 18 i + 18 i + 6 (i^{1} i)}{(9 - 3 i) (9 + 3 i)}$

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

$\frac{54 + 18 i + 18 i + 6 (i^{1} i^{1})}{(9 - 3 i) (9 + 3 i)}$

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

$\frac{54 + 18 i + 18 i + 6 i^{1 + 1}}{(9 - 3 i) (9 + 3 i)}$

Add $1$ and $1$ .

$\frac{54 + 18 i + 18 i + 6 i^{2}}{(9 - 3 i) (9 + 3 i)}$

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

$\frac{54 + 18 i + 18 i + 6 \cdot - 1}{(9 - 3 i) (9 + 3 i)}$

Multiply $6$ by $- 1$ .

$\frac{54 + 18 i + 18 i - 6}{(9 - 3 i) (9 + 3 i)}$

Subtract $6$ from $54$ .

$\frac{48 + 18 i + 18 i}{(9 - 3 i) (9 + 3 i)}$

Add $18 i$ and $18 i$ .

$\frac{48 + 36 i}{(9 - 3 i) (9 + 3 i)}$

Simplify the denominator.

Expand $(9 - 3 i) (9 + 3 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{48 + 36 i}{9 (9 + 3 i) - 3 i (9 + 3 i)}$

Apply the distributive property.

$\frac{48 + 36 i}{9 \cdot 9 + 9 (3 i) - 3 i (9 + 3 i)}$

Apply the distributive property.

$\frac{48 + 36 i}{9 \cdot 9 + 9 (3 i) - 3 i \cdot 9 - 3 i (3 i)}$

Simplify.

Multiply $9$ by $9$ .

$\frac{48 + 36 i}{81 + 9 (3 i) - 3 i \cdot 9 - 3 i (3 i)}$

Multiply $3$ by $9$ .

$\frac{48 + 36 i}{81 + 27 i - 3 i \cdot 9 - 3 i (3 i)}$

Multiply $9$ by $- 3$ .

$\frac{48 + 36 i}{81 + 27 i - 27 i - 3 i (3 i)}$

Multiply $3$ by $- 3$ .

$\frac{48 + 36 i}{81 + 27 i - 27 i - 9 i i}$

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

$\frac{48 + 36 i}{81 + 27 i - 27 i - 9 (i^{1} i)}$

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

$\frac{48 + 36 i}{81 + 27 i - 27 i - 9 (i^{1} i^{1})}$

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

$\frac{48 + 36 i}{81 + 27 i - 27 i - 9 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{48 + 36 i}{81 + 27 i - 27 i - 9 i^{2}}$

Subtract $27 i$ from $27 i$ .

$\frac{48 + 36 i}{81 + 0 - 9 i^{2}}$

Add $81$ and $0$ .

$\frac{48 + 36 i}{81 - 9 i^{2}}$

Simplify each term.

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

$\frac{48 + 36 i}{81 - 9 \cdot - 1}$

Multiply $- 9$ by $- 1$ .

$\frac{48 + 36 i}{81 + 9}$

Add $81$ and $9$ .

$\frac{48 + 36 i}{90}$

Step 3

Cancel the common factor of $48 + 36 i$ and $90$ .

Factor $6$ out of $48$ .

$\frac{6 (8) + 36 i}{90}$

Factor $6$ out of $36 i$ .

$\frac{6 (8) + 6 (6 i)}{90}$

Factor $6$ out of $6 (8) + 6 (6 i)$ .

$\frac{6 (8 + 6 i)}{90}$

Cancel the common factors.

Factor $6$ out of $90$ .

$\frac{6 (8 + 6 i)}{6 \cdot 15}$

Cancel the common factor.

$\frac{6 (8 + 6 i)}{6 \cdot 15}$

Rewrite the expression.

$\frac{8 + 6 i}{15}$

Step 4

Split the fraction $\frac{8 + 6 i}{15}$ into two fractions.

$\frac{8}{15} + \frac{6 i}{15}$

Step 5

Cancel the common factor of $6$ and $15$ .

Factor $3$ out of $6 i$ .

$\frac{8}{15} + \frac{3 (2 i)}{15}$

Cancel the common factors.

Factor $3$ out of $15$ .

$\frac{8}{15} + \frac{3 (2 i)}{3 (5)}$

Cancel the common factor.

$\frac{8}{15} + \frac{3 (2 i)}{3 \cdot 5}$

Rewrite the expression.

$\frac{8}{15} + \frac{2 i}{5}$

Write in Standard Form (6+2i)/(9-3i)

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