Algebra Examples

\frac{- 3}{- 4 + i}

$\frac{- 3}{- 4 + i}$

Step 1

Multiply the numerator and denominator of

\frac{- 3}{- 4 + 1 i}

$\frac{- 3}{- 4 + 1 i}$ by the conjugate of

- 4 + 1 i

$- 4 + 1 i$ to make the denominator real.

\frac{- 3}{- 4 + 1 i} \cdot \frac{- 4 - i}{- 4 - i}

$\frac{- 3}{- 4 + 1 i} \cdot \frac{- 4 - i}{- 4 - i}$

Step 2

Multiply.

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Step 2.1

Combine.

\frac{- 3 (- 4 - i)}{(- 4 + 1 i) (- 4 - i)}

$\frac{- 3 (- 4 - i)}{(- 4 + 1 i) (- 4 - i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the distributive property.

\frac{- 3 \cdot - 4 - 3 (- i)}{(- 4 + 1 i) (- 4 - i)}

$\frac{- 3 \cdot - 4 - 3 (- i)}{(- 4 + 1 i) (- 4 - i)}$

Step 2.2.2

Multiply

- 3

$- 3$ by

- 4

$- 4$ .

\frac{12 - 3 (- i)}{(- 4 + 1 i) (- 4 - i)}

$\frac{12 - 3 (- i)}{(- 4 + 1 i) (- 4 - i)}$

Step 2.2.3

Multiply

- 1

$- 1$ by

- 3

$- 3$ .

\frac{12 + 3 i}{(- 4 + 1 i) (- 4 - i)}

$\frac{12 + 3 i}{(- 4 + 1 i) (- 4 - i)}$

\frac{12 + 3 i}{(- 4 + 1 i) (- 4 - i)}

$\frac{12 + 3 i}{(- 4 + 1 i) (- 4 - i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand

(- 4 + 1 i) (- 4 - i)

$(- 4 + 1 i) (- 4 - i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

\frac{12 + 3 i}{- 4 (- 4 - i) + 1 i (- 4 - i)}

$\frac{12 + 3 i}{- 4 (- 4 - i) + 1 i (- 4 - i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{12 + 3 i}{- 4 \cdot - 4 - 4 (- i) + 1 i (- 4 - i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{12 + 3 i}{- 4 \cdot - 4 - 4 (- i) + 1 i \cdot - 4 + 1 i (- i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply

$- 4$ by

$- 4$ .

$\frac{12 + 3 i}{16 - 4 (- i) + 1 i \cdot - 4 + 1 i (- i)}$

Step 2.3.2.2

Multiply

$- 1$ by

$- 4$ .

$\frac{12 + 3 i}{16 + 4 i + 1 i \cdot - 4 + 1 i (- i)}$

Step 2.3.2.3

Multiply

$- 4$ by

$1$ .

$\frac{12 + 3 i}{16 + 4 i - 4 i + 1 i (- i)}$

Step 2.3.2.4

Multiply

$- 1$ by

$1$ .

$\frac{12 + 3 i}{16 + 4 i - 4 i - i i}$

Step 2.3.2.5

Raise

$i$ to the power of

$1$ .

$\frac{12 + 3 i}{16 + 4 i - 4 i - (i^{1} i)}$

Step 2.3.2.6

Raise

$i$ to the power of

$1$ .

$\frac{12 + 3 i}{16 + 4 i - 4 i - (i^{1} i^{1})}$

Step 2.3.2.7

Use the power rule

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

$\frac{12 + 3 i}{16 + 4 i - 4 i - i^{1 + 1}}$

Step 2.3.2.8

Add

$1$ and

$1$ .

$\frac{12 + 3 i}{16 + 4 i - 4 i - i^{2}}$

Step 2.3.2.9

Subtract

$4 i$ from

$4 i$ .

$\frac{12 + 3 i}{16 + 0 - i^{2}}$

Step 2.3.2.10

Add

$16$ and

$0$ .

$\frac{12 + 3 i}{16 - i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

Rewrite

$i^{2}$ as

$- 1$ .

$\frac{12 + 3 i}{16 - - 1}$

Step 2.3.3.2

Multiply

$- 1$ by

$- 1$ .

$\frac{12 + 3 i}{16 + 1}$

Step 2.3.4

Add

$16$ and

$1$ .

$\frac{12 + 3 i}{17}$

Step 3

Split the fraction

$\frac{12 + 3 i}{17}$ into two fractions.

$\frac{12}{17} + \frac{3 i}{17}$

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