Linear Algebra Examples

$\frac{- 2}{4 - i}$

Step 1

Multiply the numerator and denominator of $\frac{- 2}{4 - i}$ by the conjugate of $4 - i$ to make the denominator real.

$\frac{- 2}{4 - i} \cdot \frac{4 + i}{4 + i}$

Step 2

Multiply.

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Step 2.1

Combine.

$\frac{- 2 (4 + i)}{(4 - i) (4 + i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Multiply $- 2$ by $4$ .

$\frac{- 8 - 2 i}{(4 - i) (4 + i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

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

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Step 2.3.1.1

Apply the distributive property.

$\frac{- 8 - 2 i}{4 (4 + i) - i (4 + i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{- 8 - 2 i}{4 \cdot 4 + 4 i - i (4 + i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{- 8 - 2 i}{4 \cdot 4 + 4 i - i \cdot 4 - i i}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $4$ by $4$ .

$\frac{- 8 - 2 i}{16 + 4 i - i \cdot 4 - i i}$

Step 2.3.2.2

Multiply $4$ by $- 1$ .

$\frac{- 8 - 2 i}{16 + 4 i - 4 i - i i}$

Step 2.3.2.3

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

$\frac{- 8 - 2 i}{16 + 4 i - 4 i - (i^{1} i)}$

Step 2.3.2.4

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

$\frac{- 8 - 2 i}{16 + 4 i - 4 i - (i^{1} i^{1})}$

Step 2.3.2.5

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

$\frac{- 8 - 2 i}{16 + 4 i - 4 i - i^{1 + 1}}$

Step 2.3.2.6

Add $1$ and $1$ .

$\frac{- 8 - 2 i}{16 + 4 i - 4 i - i^{2}}$

Step 2.3.2.7

Subtract $4 i$ from $4 i$ .

$\frac{- 8 - 2 i}{16 + 0 - i^{2}}$

Step 2.3.2.8

Add $16$ and $0$ .

$\frac{- 8 - 2 i}{16 - i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{- 8 - 2 i}{16 - - 1}$

Step 2.3.3.2

Multiply $- 1$ by $- 1$ .

$\frac{- 8 - 2 i}{16 + 1}$

Step 2.3.4

Add $16$ and $1$ .

$\frac{- 8 - 2 i}{17}$

Step 3

Split the fraction $\frac{- 8 - 2 i}{17}$ into two fractions.

$\frac{- 8}{17} + \frac{- 2 i}{17}$

Step 4

Simplify each term.

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Step 4.1

Move the negative in front of the fraction.

$- \frac{8}{17} + \frac{- 2 i}{17}$

Step 4.2

Move the negative in front of the fraction.

$- \frac{8}{17} - \frac{2 i}{17}$

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