Algebra Examples

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

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.

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Step 2.1

Combine.

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

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

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Step 2.2.1.1

Apply the distributive property.

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

Step 2.2.1.2

Apply the distributive property.

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

Step 2.2.1.3

Apply the distributive property.

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

Step 2.2.2

Simplify and combine like terms.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Multiply $6$ by $9$ .

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

Step 2.2.2.1.2

Multiply $3$ by $6$ .

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

Step 2.2.2.1.3

Multiply $9$ by $2$ .

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

Step 2.2.2.1.4

Multiply $2 i (3 i)$ .

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Step 2.2.2.1.4.1

Multiply $3$ by $2$ .

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

Step 2.2.2.1.4.2

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

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

Step 2.2.2.1.4.3

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

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

Step 2.2.2.1.4.4

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

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

Step 2.2.2.1.4.5

Add $1$ and $1$ .

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

Step 2.2.2.1.5

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

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

Step 2.2.2.1.6

Multiply $6$ by $- 1$ .

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

Step 2.2.2.2

Subtract $6$ from $54$ .

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

Step 2.2.2.3

Add $18 i$ and $18 i$ .

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

Step 2.3

Simplify the denominator.

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Step 2.3.1

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

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Step 2.3.1.1

Apply the distributive property.

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

Step 2.3.1.2

Apply the distributive property.

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

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $9$ by $9$ .

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

Step 2.3.2.2

Multiply $3$ by $9$ .

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

Step 2.3.2.3

Multiply $9$ by $- 3$ .

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

Step 2.3.2.4

Multiply $3$ by $- 3$ .

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

Step 2.3.2.5

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

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

Step 2.3.2.6

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

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

Step 2.3.2.7

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

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

Step 2.3.2.8

Add $1$ and $1$ .

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

Step 2.3.2.9

Subtract $27 i$ from $27 i$ .

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

Step 2.3.2.10

Add $81$ and $0$ .

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

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

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

Step 2.3.3.2

Multiply $- 9$ by $- 1$ .

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

Step 2.3.4

Add $81$ and $9$ .

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

Step 3

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

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Step 3.1

Factor $6$ out of $48$ .

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

Step 3.2

Factor $6$ out of $36 i$ .

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

Step 3.3

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

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

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor $6$ out of $90$ .

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

Step 3.4.2

Cancel the common factor.

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

Step 3.4.3

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 $15$ .

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Step 5.1

Factor $3$ out of $6 i$ .

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

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $3$ out of $15$ .

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

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