Linear Algebra Examples

$\frac{- 7}{2 - 9 i}$

Step 1

Multiply the numerator and denominator of $\frac{- 7}{2 - 9 i}$ by the conjugate of $2 - 9 i$ to make the denominator real.

$\frac{- 7}{2 - 9 i} \cdot \frac{2 + 9 i}{2 + 9 i}$

Step 2

Multiply.

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Step 2.1

Combine.

$\frac{- 7 (2 + 9 i)}{(2 - 9 i) (2 + 9 i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the distributive property.

$\frac{- 7 \cdot 2 - 7 (9 i)}{(2 - 9 i) (2 + 9 i)}$

Step 2.2.2

Multiply $- 7$ by $2$ .

$\frac{- 14 - 7 (9 i)}{(2 - 9 i) (2 + 9 i)}$

Step 2.2.3

Multiply $9$ by $- 7$ .

$\frac{- 14 - 63 i}{(2 - 9 i) (2 + 9 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand $(2 - 9 i) (2 + 9 i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

$\frac{- 14 - 63 i}{2 (2 + 9 i) - 9 i (2 + 9 i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{- 14 - 63 i}{2 \cdot 2 + 2 (9 i) - 9 i (2 + 9 i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{- 14 - 63 i}{2 \cdot 2 + 2 (9 i) - 9 i \cdot 2 - 9 i (9 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $2$ by $2$ .

$\frac{- 14 - 63 i}{4 + 2 (9 i) - 9 i \cdot 2 - 9 i (9 i)}$

Step 2.3.2.2

Multiply $9$ by $2$ .

$\frac{- 14 - 63 i}{4 + 18 i - 9 i \cdot 2 - 9 i (9 i)}$

Step 2.3.2.3

Multiply $2$ by $- 9$ .

$\frac{- 14 - 63 i}{4 + 18 i - 18 i - 9 i (9 i)}$

Step 2.3.2.4

Multiply $9$ by $- 9$ .

$\frac{- 14 - 63 i}{4 + 18 i - 18 i - 81 i i}$

Step 2.3.2.5

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

$\frac{- 14 - 63 i}{4 + 18 i - 18 i - 81 (i^{1} i)}$

Step 2.3.2.6

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

$\frac{- 14 - 63 i}{4 + 18 i - 18 i - 81 (i^{1} i^{1})}$

Step 2.3.2.7

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

$\frac{- 14 - 63 i}{4 + 18 i - 18 i - 81 i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{- 14 - 63 i}{4 + 18 i - 18 i - 81 i^{2}}$

Step 2.3.2.9

Subtract $18 i$ from $18 i$ .

$\frac{- 14 - 63 i}{4 + 0 - 81 i^{2}}$

Step 2.3.2.10

Add $4$ and $0$ .

$\frac{- 14 - 63 i}{4 - 81 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{- 14 - 63 i}{4 - 81 \cdot - 1}$

Step 2.3.3.2

Multiply $- 81$ by $- 1$ .

$\frac{- 14 - 63 i}{4 + 81}$

Step 2.3.4

Add $4$ and $81$ .

$\frac{- 14 - 63 i}{85}$

Step 3

Split the fraction $\frac{- 14 - 63 i}{85}$ into two fractions.

$\frac{- 14}{85} + \frac{- 63 i}{85}$

Step 4

Simplify each term.

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Step 4.1

Move the negative in front of the fraction.

$- \frac{14}{85} + \frac{- 63 i}{85}$

Step 4.2

Move the negative in front of the fraction.

$- \frac{14}{85} - \frac{63 i}{85}$

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