Examples

\frac{5 i + 15}{2 i - 1}

$\frac{5 i + 15}{2 i - 1}$

Step 1

Multiply the numerator and denominator of

\frac{5 i + 15}{- 1 + 2 i}

$\frac{5 i + 15}{- 1 + 2 i}$ by the conjugate of

- 1 + 2 i

$- 1 + 2 i$ to make the denominator real.

\frac{5 i + 15}{- 1 + 2 i} \cdot \frac{- 1 - 2 i}{- 1 - 2 i}

$\frac{5 i + 15}{- 1 + 2 i} \cdot \frac{- 1 - 2 i}{- 1 - 2 i}$

Step 2

Multiply.

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Step 2.1

Combine.

\frac{(5 i + 15) (- 1 - 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}

$\frac{(5 i + 15) (- 1 - 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand

(5 i + 15) (- 1 - 2 i)

$(5 i + 15) (- 1 - 2 i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

\frac{5 i (- 1 - 2 i) + 15 (- 1 - 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}

$\frac{5 i (- 1 - 2 i) + 15 (- 1 - 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}$

Step 2.2.1.2

Apply the distributive property.

\frac{5 i \cdot - 1 + 5 i (- 2 i) + 15 (- 1 - 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}

$\frac{5 i \cdot - 1 + 5 i (- 2 i) + 15 (- 1 - 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}$

Step 2.2.1.3

Apply the distributive property.

\frac{5 i \cdot - 1 + 5 i (- 2 i) + 15 \cdot - 1 + 15 (- 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}

$\frac{5 i \cdot - 1 + 5 i (- 2 i) + 15 \cdot - 1 + 15 (- 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}$

\frac{5 i \cdot - 1 + 5 i (- 2 i) + 15 \cdot - 1 + 15 (- 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}

$\frac{5 i \cdot - 1 + 5 i (- 2 i) + 15 \cdot - 1 + 15 (- 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}$

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

- 1

$- 1$ by

5

$5$ .

\frac{- 5 i + 5 i (- 2 i) + 15 \cdot - 1 + 15 (- 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}

$\frac{- 5 i + 5 i (- 2 i) + 15 \cdot - 1 + 15 (- 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}$

Step 2.2.2.1.2

Multiply

5 i (- 2 i)

$5 i (- 2 i)$ .

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Step 2.2.2.1.2.1

Multiply

- 2

$- 2$ by

5

$5$ .

\frac{- 5 i - 10 i i + 15 \cdot - 1 + 15 (- 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}

$\frac{- 5 i - 10 i i + 15 \cdot - 1 + 15 (- 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}$

Step 2.2.2.1.2.2

Raise

i

$i$ to the power of

1

$1$ .

\frac{- 5 i - 10 (i^{1} i) + 15 \cdot - 1 + 15 (- 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}

$\frac{- 5 i - 10 (i^{1} i) + 15 \cdot - 1 + 15 (- 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}$

Step 2.2.2.1.2.3

Raise

i

$i$ to the power of

1

$1$ .

\frac{- 5 i - 10 (i^{1} i^{1}) + 15 \cdot - 1 + 15 (- 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}

$\frac{- 5 i - 10 (i^{1} i^{1}) + 15 \cdot - 1 + 15 (- 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}$

Step 2.2.2.1.2.4

Use the power rule

a^{m} a^{n} = a^{m + n}

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

\frac{- 5 i - 10 i^{1 + 1} + 15 \cdot - 1 + 15 (- 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}

$\frac{- 5 i - 10 i^{1 + 1} + 15 \cdot - 1 + 15 (- 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}$

Step 2.2.2.1.2.5

Add

1

$1$ and

1

$1$ .

\frac{- 5 i - 10 i^{2} + 15 \cdot - 1 + 15 (- 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}

$\frac{- 5 i - 10 i^{2} + 15 \cdot - 1 + 15 (- 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}$

\frac{- 5 i - 10 i^{2} + 15 \cdot - 1 + 15 (- 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}

$\frac{- 5 i - 10 i^{2} + 15 \cdot - 1 + 15 (- 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}$

Step 2.2.2.1.3

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

\frac{- 5 i - 10 \cdot - 1 + 15 \cdot - 1 + 15 (- 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}

$\frac{- 5 i - 10 \cdot - 1 + 15 \cdot - 1 + 15 (- 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}$

Step 2.2.2.1.4

Multiply

- 10

$- 10$ by

- 1

$- 1$ .

\frac{- 5 i + 10 + 15 \cdot - 1 + 15 (- 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}

$\frac{- 5 i + 10 + 15 \cdot - 1 + 15 (- 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}$

Step 2.2.2.1.5

Multiply

$15$ by

$- 1$ .

$\frac{- 5 i + 10 - 15 + 15 (- 2 i)}{(- 1 + 2 i) (- 1 - 2 i)}$

Step 2.2.2.1.6

Multiply

$- 2$ by

$15$ .

$\frac{- 5 i + 10 - 15 - 30 i}{(- 1 + 2 i) (- 1 - 2 i)}$

Step 2.2.2.2

Subtract

$30 i$ from

$- 5 i$ .

$\frac{10 - 15 - 35 i}{(- 1 + 2 i) (- 1 - 2 i)}$

Step 2.2.2.3

Subtract

$15$ from

$10$ .

$\frac{- 5 - 35 i}{(- 1 + 2 i) (- 1 - 2 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand

$(- 1 + 2 i) (- 1 - 2 i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

$\frac{- 5 - 35 i}{- 1 (- 1 - 2 i) + 2 i (- 1 - 2 i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{- 5 - 35 i}{- 1 \cdot - 1 - 1 (- 2 i) + 2 i (- 1 - 2 i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{- 5 - 35 i}{- 1 \cdot - 1 - 1 (- 2 i) + 2 i \cdot - 1 + 2 i (- 2 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply

$- 1$ by

$- 1$ .

$\frac{- 5 - 35 i}{1 - 1 (- 2 i) + 2 i \cdot - 1 + 2 i (- 2 i)}$

Step 2.3.2.2

Multiply

$- 2$ by

$- 1$ .

$\frac{- 5 - 35 i}{1 + 2 i + 2 i \cdot - 1 + 2 i (- 2 i)}$

Step 2.3.2.3

Multiply

$- 1$ by

$2$ .

$\frac{- 5 - 35 i}{1 + 2 i - 2 i + 2 i (- 2 i)}$

Step 2.3.2.4

Multiply

$- 2$ by

$2$ .

$\frac{- 5 - 35 i}{1 + 2 i - 2 i - 4 i i}$

Step 2.3.2.5

Raise

$i$ to the power of

$1$ .

$\frac{- 5 - 35 i}{1 + 2 i - 2 i - 4 (i^{1} i)}$

Step 2.3.2.6

Raise

$i$ to the power of

$1$ .

$\frac{- 5 - 35 i}{1 + 2 i - 2 i - 4 (i^{1} i^{1})}$

Step 2.3.2.7

Use the power rule

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

$\frac{- 5 - 35 i}{1 + 2 i - 2 i - 4 i^{1 + 1}}$

Step 2.3.2.8

Add

$1$ and

$1$ .

$\frac{- 5 - 35 i}{1 + 2 i - 2 i - 4 i^{2}}$

Step 2.3.2.9

Subtract

$2 i$ from

$2 i$ .

$\frac{- 5 - 35 i}{1 + 0 - 4 i^{2}}$

Step 2.3.2.10

Add

$1$ and

$0$ .

$\frac{- 5 - 35 i}{1 - 4 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

Rewrite

$i^{2}$ as

$- 1$ .

$\frac{- 5 - 35 i}{1 - 4 \cdot - 1}$

Step 2.3.3.2

Multiply

$- 4$ by

$- 1$ .

$\frac{- 5 - 35 i}{1 + 4}$

Step 2.3.4

Add

$1$ and

$4$ .

$\frac{- 5 - 35 i}{5}$

Step 3

Cancel the common factor of

$- 5 - 35 i$ and

$5$ .

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Step 3.1

Factor

$5$ out of

$- 5$ .

$\frac{5 \cdot - 1 - 35 i}{5}$

Step 3.2

Factor

$5$ out of

$- 35 i$ .

$\frac{5 \cdot - 1 + 5 (- 7 i)}{5}$

Step 3.3

Factor

$5$ out of

$5 (- 1) + 5 (- 7 i)$ .

$\frac{5 (- 1 - 7 i)}{5}$

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor

$5$ out of

$5$ .

$\frac{5 (- 1 - 7 i)}{5 (1)}$

Step 3.4.2

Cancel the common factor.

$\frac{5 (- 1 - 7 i)}{5 \cdot 1}$

Step 3.4.3

Rewrite the expression.

$\frac{- 1 - 7 i}{1}$

Step 3.4.4

Divide

$- 1 - 7 i$ by

$1$ .

$- 1 - 7 i$

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