Basic Math Examples

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

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

Step 1

Cancel the common factor of

3 + 9 i

$3 + 9 i$ and

3 - 9 i

$3 - 9 i$ .

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Step 1.1

Factor

3

$3$ out of

3

$3$ .

\frac{3 \cdot 1 + 9 i}{3 - 9 i}

$\frac{3 \cdot 1 + 9 i}{3 - 9 i}$

Step 1.2

Factor

3

$3$ out of

9 i

$9 i$ .

\frac{3 \cdot 1 + 3 (3 i)}{3 - 9 i}

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

Step 1.3

Factor

3

$3$ out of

3 (1) + 3 (3 i)

$3 (1) + 3 (3 i)$ .

\frac{3 (1 + 3 i)}{3 - 9 i}

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

Step 1.4

Cancel the common factors.

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Step 1.4.1

Factor

3

$3$ out of

3

$3$ .

\frac{3 (1 + 3 i)}{3 (1) - 9 i}

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

Step 1.4.2

Factor

3

$3$ out of

- 9 i

$- 9 i$ .

\frac{3 (1 + 3 i)}{3 (1) + 3 (- 3 i)}

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

Step 1.4.3

Factor

3

$3$ out of

3 (1) + 3 (- 3 i)

$3 (1) + 3 (- 3 i)$ .

\frac{3 (1 + 3 i)}{3 (1 - 3 i)}

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

Step 1.4.4

Cancel the common factor.

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

Step 1.4.5

Rewrite the expression.

$\frac{1 + 3 i}{1 - 3 i}$

Step 2

Multiply the numerator and denominator of

$\frac{1 + 3 i}{1 - 3 i}$ by the conjugate of

$1 - 3 i$ to make the denominator real.

$\frac{1 + 3 i}{1 - 3 i} \cdot \frac{1 + 3 i}{1 + 3 i}$

Step 3

Multiply.

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Step 3.1

Combine.

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

Step 3.2

Simplify the numerator.

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Step 3.2.1

Expand

$(1 + 3 i) (1 + 3 i)$ using the FOIL Method.

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Step 3.2.1.1

Apply the distributive property.

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

Step 3.2.1.2

Apply the distributive property.

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

Step 3.2.1.3

Apply the distributive property.

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

Step 3.2.2

Simplify and combine like terms.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Multiply

$1$ by

$1$ .

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

Step 3.2.2.1.2

Multiply

$3 i$ by

$1$ .

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

Step 3.2.2.1.3

Multiply

$3$ by

$1$ .

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

Step 3.2.2.1.4

Multiply

$3 i (3 i)$ .

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Step 3.2.2.1.4.1

Multiply

$3$ by

$3$ .

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

Step 3.2.2.1.4.2

Raise

$i$ to the power of

$1$ .

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

Step 3.2.2.1.4.3

Raise

$i$ to the power of

$1$ .

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

Step 3.2.2.1.4.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.2.2.1.4.5

Add

$1$ and

$1$ .

$\frac{1 + 3 i + 3 i + 9 i^{2}}{(1 - 3 i) (1 + 3 i)}$

Step 3.2.2.1.5

Rewrite

$i^{2}$ as

$- 1$ .

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

Step 3.2.2.1.6

Multiply

$9$ by

$- 1$ .

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

Step 3.2.2.2

Subtract

$9$ from

$1$ .

$\frac{- 8 + 3 i + 3 i}{(1 - 3 i) (1 + 3 i)}$

Step 3.2.2.3

Add

$3 i$ and

$3 i$ .

$\frac{- 8 + 6 i}{(1 - 3 i) (1 + 3 i)}$

Step 3.3

Simplify the denominator.

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Step 3.3.1

Expand

$(1 - 3 i) (1 + 3 i)$ using the FOIL Method.

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Step 3.3.1.1

Apply the distributive property.

$\frac{- 8 + 6 i}{1 (1 + 3 i) - 3 i (1 + 3 i)}$

Step 3.3.1.2

Apply the distributive property.

$\frac{- 8 + 6 i}{1 \cdot 1 + 1 (3 i) - 3 i (1 + 3 i)}$

Step 3.3.1.3

Apply the distributive property.

$\frac{- 8 + 6 i}{1 \cdot 1 + 1 (3 i) - 3 i \cdot 1 - 3 i (3 i)}$

Step 3.3.2

Simplify.

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Step 3.3.2.1

Multiply

$1$ by

$1$ .

$\frac{- 8 + 6 i}{1 + 1 (3 i) - 3 i \cdot 1 - 3 i (3 i)}$

Step 3.3.2.2

Multiply

$3$ by

$1$ .

$\frac{- 8 + 6 i}{1 + 3 i - 3 i \cdot 1 - 3 i (3 i)}$

Step 3.3.2.3

Multiply

$- 3$ by

$1$ .

$\frac{- 8 + 6 i}{1 + 3 i - 3 i - 3 i (3 i)}$

Step 3.3.2.4

Multiply

$3$ by

$- 3$ .

$\frac{- 8 + 6 i}{1 + 3 i - 3 i - 9 i i}$

Step 3.3.2.5

Raise

$i$ to the power of

$1$ .

$\frac{- 8 + 6 i}{1 + 3 i - 3 i - 9 (i^{1} i)}$

Step 3.3.2.6

Raise

$i$ to the power of

$1$ .

$\frac{- 8 + 6 i}{1 + 3 i - 3 i - 9 (i^{1} i^{1})}$

Step 3.3.2.7

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{- 8 + 6 i}{1 + 3 i - 3 i - 9 i^{1 + 1}}$

Step 3.3.2.8

Add

$1$ and

$1$ .

$\frac{- 8 + 6 i}{1 + 3 i - 3 i - 9 i^{2}}$

Step 3.3.2.9

Subtract

$3 i$ from

$3 i$ .

$\frac{- 8 + 6 i}{1 + 0 - 9 i^{2}}$

Step 3.3.2.10

Add

$1$ and

$0$ .

$\frac{- 8 + 6 i}{1 - 9 i^{2}}$

Step 3.3.3

Simplify each term.

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Step 3.3.3.1

Rewrite

$i^{2}$ as

$- 1$ .

$\frac{- 8 + 6 i}{1 - 9 \cdot - 1}$

Step 3.3.3.2

Multiply

$- 9$ by

$- 1$ .

$\frac{- 8 + 6 i}{1 + 9}$

Step 3.3.4

Add

$1$ and

$9$ .

$\frac{- 8 + 6 i}{10}$

Step 4

Cancel the common factor of

$- 8 + 6 i$ and

$10$ .

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Step 4.1

Factor

$2$ out of

$- 8$ .

$\frac{2 (- 4) + 6 i}{10}$

Step 4.2

Factor

$2$ out of

$6 i$ .

$\frac{2 (- 4) + 2 (3 i)}{10}$

Step 4.3

Factor

$2$ out of

$2 (- 4) + 2 (3 i)$ .

$\frac{2 (- 4 + 3 i)}{10}$

Step 4.4

Cancel the common factors.

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Step 4.4.1

Factor

$2$ out of

$10$ .

$\frac{2 (- 4 + 3 i)}{2 \cdot 5}$

Step 4.4.2

Cancel the common factor.

$\frac{2 (- 4 + 3 i)}{2 \cdot 5}$

Step 4.4.3

Rewrite the expression.

$\frac{- 4 + 3 i}{5}$

Step 5

Split the fraction

$\frac{- 4 + 3 i}{5}$ into two fractions.

$\frac{- 4}{5} + \frac{3 i}{5}$

Step 6

Move the negative in front of the fraction.

$- \frac{4}{5} + \frac{3 i}{5}$