Examples

\frac{12 i + 1}{i - 1}

$\frac{12 i + 1}{i - 1}$

Step 1

Multiply the numerator and denominator of

\frac{12 i + 1}{- 1 + 1 i}

$\frac{12 i + 1}{- 1 + 1 i}$ by the conjugate of

- 1 + 1 i

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

\frac{12 i + 1}{- 1 + 1 i} \cdot \frac{- 1 - i}{- 1 - i}

$\frac{12 i + 1}{- 1 + 1 i} \cdot \frac{- 1 - i}{- 1 - i}$

Step 2

Multiply.

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Step 2.1

Combine.

\frac{(12 i + 1) (- 1 - i)}{(- 1 + 1 i) (- 1 - i)}

$\frac{(12 i + 1) (- 1 - i)}{(- 1 + 1 i) (- 1 - i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand

(12 i + 1) (- 1 - i)

$(12 i + 1) (- 1 - i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

\frac{12 i (- 1 - i) + 1 (- 1 - i)}{(- 1 + 1 i) (- 1 - i)}

$\frac{12 i (- 1 - i) + 1 (- 1 - i)}{(- 1 + 1 i) (- 1 - i)}$

Step 2.2.1.2

Apply the distributive property.

\frac{12 i \cdot - 1 + 12 i (- i) + 1 (- 1 - i)}{(- 1 + 1 i) (- 1 - i)}

$\frac{12 i \cdot - 1 + 12 i (- i) + 1 (- 1 - i)}{(- 1 + 1 i) (- 1 - i)}$

Step 2.2.1.3

Apply the distributive property.

\frac{12 i \cdot - 1 + 12 i (- i) + 1 \cdot - 1 + 1 (- i)}{(- 1 + 1 i) (- 1 - i)}

$\frac{12 i \cdot - 1 + 12 i (- i) + 1 \cdot - 1 + 1 (- i)}{(- 1 + 1 i) (- 1 - i)}$

\frac{12 i \cdot - 1 + 12 i (- i) + 1 \cdot - 1 + 1 (- i)}{(- 1 + 1 i) (- 1 - i)}

$\frac{12 i \cdot - 1 + 12 i (- i) + 1 \cdot - 1 + 1 (- i)}{(- 1 + 1 i) (- 1 - i)}$

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

12

$12$ .

\frac{- 12 i + 12 i (- i) + 1 \cdot - 1 + 1 (- i)}{(- 1 + 1 i) (- 1 - i)}

$\frac{- 12 i + 12 i (- i) + 1 \cdot - 1 + 1 (- i)}{(- 1 + 1 i) (- 1 - i)}$

Step 2.2.2.1.2

Multiply

12 i (- i)

$12 i (- i)$ .

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Step 2.2.2.1.2.1

Multiply

- 1

$- 1$ by

12

$12$ .

\frac{- 12 i - 12 i i + 1 \cdot - 1 + 1 (- i)}{(- 1 + 1 i) (- 1 - i)}

$\frac{- 12 i - 12 i i + 1 \cdot - 1 + 1 (- i)}{(- 1 + 1 i) (- 1 - i)}$

Step 2.2.2.1.2.2

Raise

i

$i$ to the power of

1

$1$ .

\frac{- 12 i - 12 (i^{1} i) + 1 \cdot - 1 + 1 (- i)}{(- 1 + 1 i) (- 1 - i)}

$\frac{- 12 i - 12 (i^{1} i) + 1 \cdot - 1 + 1 (- i)}{(- 1 + 1 i) (- 1 - i)}$

Step 2.2.2.1.2.3

Raise

i

$i$ to the power of

1

$1$ .

\frac{- 12 i - 12 (i^{1} i^{1}) + 1 \cdot - 1 + 1 (- i)}{(- 1 + 1 i) (- 1 - i)}

$\frac{- 12 i - 12 (i^{1} i^{1}) + 1 \cdot - 1 + 1 (- i)}{(- 1 + 1 i) (- 1 - i)}$

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{- 12 i - 12 i^{1 + 1} + 1 \cdot - 1 + 1 (- i)}{(- 1 + 1 i) (- 1 - i)}

$\frac{- 12 i - 12 i^{1 + 1} + 1 \cdot - 1 + 1 (- i)}{(- 1 + 1 i) (- 1 - i)}$

Step 2.2.2.1.2.5

Add

1

$1$ and

1

$1$ .

\frac{- 12 i - 12 i^{2} + 1 \cdot - 1 + 1 (- i)}{(- 1 + 1 i) (- 1 - i)}

$\frac{- 12 i - 12 i^{2} + 1 \cdot - 1 + 1 (- i)}{(- 1 + 1 i) (- 1 - i)}$

\frac{- 12 i - 12 i^{2} + 1 \cdot - 1 + 1 (- i)}{(- 1 + 1 i) (- 1 - i)}

$\frac{- 12 i - 12 i^{2} + 1 \cdot - 1 + 1 (- i)}{(- 1 + 1 i) (- 1 - i)}$

Step 2.2.2.1.3

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

\frac{- 12 i - 12 \cdot - 1 + 1 \cdot - 1 + 1 (- i)}{(- 1 + 1 i) (- 1 - i)}

$\frac{- 12 i - 12 \cdot - 1 + 1 \cdot - 1 + 1 (- i)}{(- 1 + 1 i) (- 1 - i)}$

Step 2.2.2.1.4

Multiply

- 12

$- 12$ by

- 1

$- 1$ .

\frac{- 12 i + 12 + 1 \cdot - 1 + 1 (- i)}{(- 1 + 1 i) (- 1 - i)}

$\frac{- 12 i + 12 + 1 \cdot - 1 + 1 (- i)}{(- 1 + 1 i) (- 1 - i)}$

Step 2.2.2.1.5

Multiply

- 1

$- 1$ by

1

$1$ .

\frac{- 12 i + 12 - 1 + 1 (- i)}{(- 1 + 1 i) (- 1 - i)}

$\frac{- 12 i + 12 - 1 + 1 (- i)}{(- 1 + 1 i) (- 1 - i)}$

Step 2.2.2.1.6

Multiply

- i

$- i$ by

1

$1$ .

\frac{- 12 i + 12 - 1 - i}{(- 1 + 1 i) (- 1 - i)}

$\frac{- 12 i + 12 - 1 - i}{(- 1 + 1 i) (- 1 - i)}$

\frac{- 12 i + 12 - 1 - i}{(- 1 + 1 i) (- 1 - i)}

$\frac{- 12 i + 12 - 1 - i}{(- 1 + 1 i) (- 1 - i)}$

Step 2.2.2.2

Subtract

i

$i$ from

- 12 i

$- 12 i$ .

\frac{12 - 1 - 13 i}{(- 1 + 1 i) (- 1 - i)}

$\frac{12 - 1 - 13 i}{(- 1 + 1 i) (- 1 - i)}$

Step 2.2.2.3

Subtract

1

$1$ from

12

$12$ .

\frac{11 - 13 i}{(- 1 + 1 i) (- 1 - i)}

$\frac{11 - 13 i}{(- 1 + 1 i) (- 1 - i)}$

\frac{11 - 13 i}{(- 1 + 1 i) (- 1 - i)}

$\frac{11 - 13 i}{(- 1 + 1 i) (- 1 - i)}$

\frac{11 - 13 i}{(- 1 + 1 i) (- 1 - i)}

$\frac{11 - 13 i}{(- 1 + 1 i) (- 1 - i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand

(- 1 + 1 i) (- 1 - i)

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

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Step 2.3.1.1

Apply the distributive property.

\frac{11 - 13 i}{- 1 (- 1 - i) + 1 i (- 1 - i)}

$\frac{11 - 13 i}{- 1 (- 1 - i) + 1 i (- 1 - i)}$

Step 2.3.1.2

Apply the distributive property.

\frac{11 - 13 i}{- 1 \cdot - 1 - 1 (- i) + 1 i (- 1 - i)}

$\frac{11 - 13 i}{- 1 \cdot - 1 - 1 (- i) + 1 i (- 1 - i)}$

Step 2.3.1.3

Apply the distributive property.

\frac{11 - 13 i}{- 1 \cdot - 1 - 1 (- i) + 1 i \cdot - 1 + 1 i (- i)}

$\frac{11 - 13 i}{- 1 \cdot - 1 - 1 (- i) + 1 i \cdot - 1 + 1 i (- i)}$

\frac{11 - 13 i}{- 1 \cdot - 1 - 1 (- i) + 1 i \cdot - 1 + 1 i (- i)}

$\frac{11 - 13 i}{- 1 \cdot - 1 - 1 (- i) + 1 i \cdot - 1 + 1 i (- i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

\frac{11 - 13 i}{1 - 1 (- i) + 1 i \cdot - 1 + 1 i (- i)}

$\frac{11 - 13 i}{1 - 1 (- i) + 1 i \cdot - 1 + 1 i (- i)}$

Step 2.3.2.2

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

\frac{11 - 13 i}{1 + 1 i + 1 i \cdot - 1 + 1 i (- i)}

$\frac{11 - 13 i}{1 + 1 i + 1 i \cdot - 1 + 1 i (- i)}$

Step 2.3.2.3

Multiply

- 1

$- 1$ by

1

$1$ .

\frac{11 - 13 i}{1 + 1 i - i + 1 i (- i)}

$\frac{11 - 13 i}{1 + 1 i - i + 1 i (- i)}$

Step 2.3.2.4

Multiply

- 1

$- 1$ by

1

$1$ .

\frac{11 - 13 i}{1 + 1 i - i - i i}

$\frac{11 - 13 i}{1 + 1 i - i - i i}$

Step 2.3.2.5

Raise

i

$i$ to the power of

1

$1$ .

\frac{11 - 13 i}{1 + 1 i - i - (i^{1} i)}

$\frac{11 - 13 i}{1 + 1 i - i - (i^{1} i)}$

Step 2.3.2.6

Raise

i

$i$ to the power of

1

$1$ .

\frac{11 - 13 i}{1 + 1 i - i - (i^{1} i^{1})}

$\frac{11 - 13 i}{1 + 1 i - i - (i^{1} i^{1})}$

Step 2.3.2.7

Use the power rule

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

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

\frac{11 - 13 i}{1 + 1 i - i - i^{1 + 1}}

$\frac{11 - 13 i}{1 + 1 i - i - i^{1 + 1}}$

Step 2.3.2.8

Add

1

$1$ and

1

$1$ .

\frac{11 - 13 i}{1 + 1 i - i - i^{2}}

$\frac{11 - 13 i}{1 + 1 i - i - i^{2}}$

Step 2.3.2.9

Subtract

i

$i$ from

1 i

$1 i$ .

\frac{11 - 13 i}{1 + 0 - i^{2}}

$\frac{11 - 13 i}{1 + 0 - i^{2}}$

Step 2.3.2.10

Add

1

$1$ and

0

$0$ .

\frac{11 - 13 i}{1 - i^{2}}

$\frac{11 - 13 i}{1 - i^{2}}$

\frac{11 - 13 i}{1 - i^{2}}

$\frac{11 - 13 i}{1 - i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

\frac{11 - 13 i}{1 - - 1}

$\frac{11 - 13 i}{1 - - 1}$

Step 2.3.3.2

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

\frac{11 - 13 i}{1 + 1}

$\frac{11 - 13 i}{1 + 1}$

\frac{11 - 13 i}{1 + 1}

$\frac{11 - 13 i}{1 + 1}$

Step 2.3.4

Add

1

$1$ and

1

$1$ .

\frac{11 - 13 i}{2}

$\frac{11 - 13 i}{2}$

\frac{11 - 13 i}{2}

$\frac{11 - 13 i}{2}$

\frac{11 - 13 i}{2}

$\frac{11 - 13 i}{2}$

Step 3

Split the fraction

\frac{11 - 13 i}{2}

$\frac{11 - 13 i}{2}$ into two fractions.

\frac{11}{2} + \frac{- 13 i}{2}

$\frac{11}{2} + \frac{- 13 i}{2}$

Step 4

Move the negative in front of the fraction.

\frac{11}{2} - \frac{13 i}{2}

$\frac{11}{2} - \frac{13 i}{2}$

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