Examples

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

Step 1

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

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

Step 2

Multiply.

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Step 2.1

Combine.

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

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

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Step 2.2.1.1

Apply the distributive property.

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

Step 2.2.1.2

Apply the distributive property.

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

Step 2.2.1.3

Apply the distributive property.

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

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

Move $- 1$ to the left of $i$ .

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

Step 2.2.2.1.2

Rewrite $- 1 i$ as $- i$ .

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

Step 2.2.2.1.3

Multiply $i (- 5 i)$ .

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Step 2.2.2.1.3.1

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

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

Step 2.2.2.1.3.2

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

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

Step 2.2.2.1.3.3

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

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

Step 2.2.2.1.3.4

Add $1$ and $1$ .

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

Step 2.2.2.1.4

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

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

Step 2.2.2.1.5

Multiply $- 5$ by $- 1$ .

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

Step 2.2.2.1.6

Multiply $- 1$ by $1$ .

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

Step 2.2.2.1.7

Multiply $- 5 i$ by $1$ .

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

Step 2.2.2.2

Subtract $5 i$ from $- i$ .

$\frac{5 - 1 - 6 i}{(- 1 + 5 i) (- 1 - 5 i)}$

Step 2.2.2.3

Subtract $1$ from $5$ .

$\frac{4 - 6 i}{(- 1 + 5 i) (- 1 - 5 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

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

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Step 2.3.1.1

Apply the distributive property.

$\frac{4 - 6 i}{- 1 (- 1 - 5 i) + 5 i (- 1 - 5 i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{4 - 6 i}{- 1 \cdot - 1 - 1 (- 5 i) + 5 i (- 1 - 5 i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{4 - 6 i}{- 1 \cdot - 1 - 1 (- 5 i) + 5 i \cdot - 1 + 5 i (- 5 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $- 1$ by $- 1$ .

$\frac{4 - 6 i}{1 - 1 (- 5 i) + 5 i \cdot - 1 + 5 i (- 5 i)}$

Step 2.3.2.2

Multiply $- 5$ by $- 1$ .

$\frac{4 - 6 i}{1 + 5 i + 5 i \cdot - 1 + 5 i (- 5 i)}$

Step 2.3.2.3

Multiply $- 1$ by $5$ .

$\frac{4 - 6 i}{1 + 5 i - 5 i + 5 i (- 5 i)}$

Step 2.3.2.4

Multiply $- 5$ by $5$ .

$\frac{4 - 6 i}{1 + 5 i - 5 i - 25 i i}$

Step 2.3.2.5

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

$\frac{4 - 6 i}{1 + 5 i - 5 i - 25 (i^{1} i)}$

Step 2.3.2.6

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

$\frac{4 - 6 i}{1 + 5 i - 5 i - 25 (i^{1} i^{1})}$

Step 2.3.2.7

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

$\frac{4 - 6 i}{1 + 5 i - 5 i - 25 i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{4 - 6 i}{1 + 5 i - 5 i - 25 i^{2}}$

Step 2.3.2.9

Subtract $5 i$ from $5 i$ .

$\frac{4 - 6 i}{1 + 0 - 25 i^{2}}$

Step 2.3.2.10

Add $1$ and $0$ .

$\frac{4 - 6 i}{1 - 25 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{4 - 6 i}{1 - 25 \cdot - 1}$

Step 2.3.3.2

Multiply $- 25$ by $- 1$ .

$\frac{4 - 6 i}{1 + 25}$

Step 2.3.4

Add $1$ and $25$ .

$\frac{4 - 6 i}{26}$

Step 3

Cancel the common factor of $4 - 6 i$ and $26$ .

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Step 3.1

Factor $2$ out of $4$ .

$\frac{2 (2) - 6 i}{26}$

Step 3.2

Factor $2$ out of $- 6 i$ .

$\frac{2 (2) + 2 (- 3 i)}{26}$

Step 3.3

Factor $2$ out of $2 (2) + 2 (- 3 i)$ .

$\frac{2 (2 - 3 i)}{26}$

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor $2$ out of $26$ .

$\frac{2 (2 - 3 i)}{2 \cdot 13}$

Step 3.4.2

Cancel the common factor.

$\frac{2 (2 - 3 i)}{2 \cdot 13}$

Step 3.4.3

Rewrite the expression.

$\frac{2 - 3 i}{13}$

Step 4

Split the fraction $\frac{2 - 3 i}{13}$ into two fractions.

$\frac{2}{13} + \frac{- 3 i}{13}$

Step 5

Move the negative in front of the fraction.

$\frac{2}{13} - \frac{3 i}{13}$

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