Examples

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

Step 1

Multiply the numerator and denominator of $\frac{5 i + 15}{- 1 + 2 i}$ by the conjugate of $- 1 + 2 i$ to make the denominator real.

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

Step 2

Multiply.

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Step 2.1

Combine.

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

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

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Step 2.2.1.1

Apply the distributive property.

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

Step 2.2.1.2

Apply the distributive property.

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

Step 2.2.1.3

Apply the distributive property.

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

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 $- 1$ by $5$ .

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

Step 2.2.2.1.2

Multiply $5 i (- 2 i)$ .

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Step 2.2.2.1.2.1

Multiply $- 2$ by $5$ .

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

Step 2.2.2.1.2.2

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

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

Step 2.2.2.1.2.3

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

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

Step 2.2.2.1.2.4

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

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

Step 2.2.2.1.2.5

Add $1$ and $1$ .

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

Step 2.2.2.1.3

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

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

Step 2.2.2.1.4

Multiply $- 10$ by $- 1$ .

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

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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