Precalculus Examples

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

Step 1

Multiply the numerator and denominator of $\frac{12 i + 1}{- 1 + 1 i}$ by the conjugate of $- 1 + 1 i$ to make the denominator real.

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

Step 2

Multiply.

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Step 2.1

Combine.

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

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

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Step 2.2.1.1

Apply the distributive property.

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

Step 2.2.1.2

Apply the distributive property.

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

Step 2.2.1.3

Apply the distributive property.

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

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 $12$ .

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

Step 2.2.2.1.2

Multiply $12 i (- i)$ .

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Step 2.2.2.1.2.1

Multiply $- 1$ by $12$ .

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

Step 2.2.2.1.2.2

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

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

Step 2.2.2.1.2.3

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

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

Step 2.2.2.1.2.4

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

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

Step 2.2.2.1.2.5

Add $1$ and $1$ .

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

Step 2.2.2.1.3

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

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

Step 2.2.2.1.4

Multiply $- 12$ by $- 1$ .

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

Step 2.2.2.1.5

Multiply $- 1$ by $1$ .

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

Step 2.2.2.1.6

Multiply $- i$ by $1$ .

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

Step 2.2.2.2

Subtract $i$ from $- 12 i$ .

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

Step 2.2.2.3

Subtract $1$ from $12$ .

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

Step 2.3

Simplify the denominator.

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Step 2.3.1

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

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Step 2.3.1.1

Apply the distributive property.

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

Step 2.3.1.2

Apply the distributive property.

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

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $- 1$ by $- 1$ .

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

Step 2.3.2.2

Multiply $- 1$ by $- 1$ .

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

Step 2.3.2.3

Multiply $- 1$ by $1$ .

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

Step 2.3.2.4

Multiply $- 1$ by $1$ .

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

Step 2.3.2.5

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

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

Step 2.3.2.6

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

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

Step 2.3.2.7

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

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

Step 2.3.2.8

Add $1$ and $1$ .

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

Step 2.3.2.9

Subtract $i$ from $1 i$ .

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

Step 2.3.2.10

Add $1$ and $0$ .

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

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

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

Step 2.3.3.2

Multiply $- 1$ by $- 1$ .

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

Step 2.3.4

Add $1$ and $1$ .

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

Step 3

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

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

Step 4

Move the negative in front of the fraction.

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

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