Linear Algebra Examples

\frac{- 5}{3 + 2 i}

$\frac{- 5}{3 + 2 i}$

Step 1

Multiply the numerator and denominator of

\frac{- 5}{3 + 2 i}

$\frac{- 5}{3 + 2 i}$ by the conjugate of

3 + 2 i

$3 + 2 i$ to make the denominator real.

\frac{- 5}{3 + 2 i} \cdot \frac{3 - 2 i}{3 - 2 i}

$\frac{- 5}{3 + 2 i} \cdot \frac{3 - 2 i}{3 - 2 i}$

Step 2

Multiply.

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Step 2.1

Combine.

\frac{- 5 (3 - 2 i)}{(3 + 2 i) (3 - 2 i)}

$\frac{- 5 (3 - 2 i)}{(3 + 2 i) (3 - 2 i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the distributive property.

\frac{- 5 \cdot 3 - 5 (- 2 i)}{(3 + 2 i) (3 - 2 i)}

$\frac{- 5 \cdot 3 - 5 (- 2 i)}{(3 + 2 i) (3 - 2 i)}$

Step 2.2.2

Multiply

- 5

$- 5$ by

3

$3$ .

\frac{- 15 - 5 (- 2 i)}{(3 + 2 i) (3 - 2 i)}

$\frac{- 15 - 5 (- 2 i)}{(3 + 2 i) (3 - 2 i)}$

Step 2.2.3

Multiply

- 2

$- 2$ by

- 5

$- 5$ .

\frac{- 15 + 10 i}{(3 + 2 i) (3 - 2 i)}

$\frac{- 15 + 10 i}{(3 + 2 i) (3 - 2 i)}$

\frac{- 15 + 10 i}{(3 + 2 i) (3 - 2 i)}

$\frac{- 15 + 10 i}{(3 + 2 i) (3 - 2 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand

(3 + 2 i) (3 - 2 i)

$(3 + 2 i) (3 - 2 i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

\frac{- 15 + 10 i}{3 (3 - 2 i) + 2 i (3 - 2 i)}

$\frac{- 15 + 10 i}{3 (3 - 2 i) + 2 i (3 - 2 i)}$

Step 2.3.1.2

Apply the distributive property.

\frac{- 15 + 10 i}{3 \cdot 3 + 3 (- 2 i) + 2 i (3 - 2 i)}

$\frac{- 15 + 10 i}{3 \cdot 3 + 3 (- 2 i) + 2 i (3 - 2 i)}$

Step 2.3.1.3

Apply the distributive property.

\frac{- 15 + 10 i}{3 \cdot 3 + 3 (- 2 i) + 2 i \cdot 3 + 2 i (- 2 i)}

$\frac{- 15 + 10 i}{3 \cdot 3 + 3 (- 2 i) + 2 i \cdot 3 + 2 i (- 2 i)}$

\frac{- 15 + 10 i}{3 \cdot 3 + 3 (- 2 i) + 2 i \cdot 3 + 2 i (- 2 i)}

$\frac{- 15 + 10 i}{3 \cdot 3 + 3 (- 2 i) + 2 i \cdot 3 + 2 i (- 2 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply

3

$3$ by

3

$3$ .

\frac{- 15 + 10 i}{9 + 3 (- 2 i) + 2 i \cdot 3 + 2 i (- 2 i)}

$\frac{- 15 + 10 i}{9 + 3 (- 2 i) + 2 i \cdot 3 + 2 i (- 2 i)}$

Step 2.3.2.2

Multiply

- 2

$- 2$ by

3

$3$ .

\frac{- 15 + 10 i}{9 - 6 i + 2 i \cdot 3 + 2 i (- 2 i)}

$\frac{- 15 + 10 i}{9 - 6 i + 2 i \cdot 3 + 2 i (- 2 i)}$

Step 2.3.2.3

Multiply

3

$3$ by

2

$2$ .

\frac{- 15 + 10 i}{9 - 6 i + 6 i + 2 i (- 2 i)}

$\frac{- 15 + 10 i}{9 - 6 i + 6 i + 2 i (- 2 i)}$

Step 2.3.2.4

Multiply

- 2

$- 2$ by

2

$2$ .

\frac{- 15 + 10 i}{9 - 6 i + 6 i - 4 i i}

$\frac{- 15 + 10 i}{9 - 6 i + 6 i - 4 i i}$

Step 2.3.2.5

Raise

i

$i$ to the power of

1

$1$ .

\frac{- 15 + 10 i}{9 - 6 i + 6 i - 4 (i^{1} i)}

$\frac{- 15 + 10 i}{9 - 6 i + 6 i - 4 (i^{1} i)}$

Step 2.3.2.6

Raise

i

$i$ to the power of

1

$1$ .

\frac{- 15 + 10 i}{9 - 6 i + 6 i - 4 (i^{1} i^{1})}

$\frac{- 15 + 10 i}{9 - 6 i + 6 i - 4 (i^{1} i^{1})}$

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{- 15 + 10 i}{9 - 6 i + 6 i - 4 i^{1 + 1}}

$\frac{- 15 + 10 i}{9 - 6 i + 6 i - 4 i^{1 + 1}}$

Step 2.3.2.8

Add

1

$1$ and

1

$1$ .

\frac{- 15 + 10 i}{9 - 6 i + 6 i - 4 i^{2}}

$\frac{- 15 + 10 i}{9 - 6 i + 6 i - 4 i^{2}}$

Step 2.3.2.9

Add

- 6 i

$- 6 i$ and

6 i

$6 i$ .

\frac{- 15 + 10 i}{9 + 0 - 4 i^{2}}

$\frac{- 15 + 10 i}{9 + 0 - 4 i^{2}}$

Step 2.3.2.10

Add

9

$9$ and

0

$0$ .

\frac{- 15 + 10 i}{9 - 4 i^{2}}

$\frac{- 15 + 10 i}{9 - 4 i^{2}}$

\frac{- 15 + 10 i}{9 - 4 i^{2}}

$\frac{- 15 + 10 i}{9 - 4 i^{2}}$

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{- 15 + 10 i}{9 - 4 \cdot - 1}

$\frac{- 15 + 10 i}{9 - 4 \cdot - 1}$

Step 2.3.3.2

Multiply

- 4

$- 4$ by

- 1

$- 1$ .

\frac{- 15 + 10 i}{9 + 4}

$\frac{- 15 + 10 i}{9 + 4}$

\frac{- 15 + 10 i}{9 + 4}

$\frac{- 15 + 10 i}{9 + 4}$

Step 2.3.4

Add

9

$9$ and

4

$4$ .

\frac{- 15 + 10 i}{13}

$\frac{- 15 + 10 i}{13}$

\frac{- 15 + 10 i}{13}

$\frac{- 15 + 10 i}{13}$

\frac{- 15 + 10 i}{13}

$\frac{- 15 + 10 i}{13}$

Step 3

Split the fraction

\frac{- 15 + 10 i}{13}

$\frac{- 15 + 10 i}{13}$ into two fractions.

\frac{- 15}{13} + \frac{10 i}{13}

$\frac{- 15}{13} + \frac{10 i}{13}$

Step 4

Move the negative in front of the fraction.

- \frac{15}{13} + \frac{10 i}{13}

$- \frac{15}{13} + \frac{10 i}{13}$

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