Algebra Examples

$\frac{6 - 2 i}{4 + i}$

Step 1

Multiply the numerator and denominator of $\frac{6 - 2 i}{4 + 1 i}$ by the conjugate of $4 + 1 i$ to make the denominator real.

$\frac{6 - 2 i}{4 + 1 i} \cdot \frac{4 - i}{4 - i}$

Step 2

Multiply.

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Step 2.1

Combine.

$\frac{(6 - 2 i) (4 - i)}{(4 + 1 i) (4 - i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand $(6 - 2 i) (4 - i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

$\frac{6 (4 - i) - 2 i (4 - i)}{(4 + 1 i) (4 - i)}$

Step 2.2.1.2

Apply the distributive property.

$\frac{6 \cdot 4 + 6 (- i) - 2 i (4 - i)}{(4 + 1 i) (4 - i)}$

Step 2.2.1.3

Apply the distributive property.

$\frac{6 \cdot 4 + 6 (- i) - 2 i \cdot 4 - 2 i (- i)}{(4 + 1 i) (4 - i)}$

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 $6$ by $4$ .

$\frac{24 + 6 (- i) - 2 i \cdot 4 - 2 i (- i)}{(4 + 1 i) (4 - i)}$

Step 2.2.2.1.2

Multiply $- 1$ by $6$ .

$\frac{24 - 6 i - 2 i \cdot 4 - 2 i (- i)}{(4 + 1 i) (4 - i)}$

Step 2.2.2.1.3

Multiply $4$ by $- 2$ .

$\frac{24 - 6 i - 8 i - 2 i (- i)}{(4 + 1 i) (4 - i)}$

Step 2.2.2.1.4

Multiply $- 2 i (- i)$ .

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Step 2.2.2.1.4.1

Multiply $- 1$ by $- 2$ .

$\frac{24 - 6 i - 8 i + 2 i i}{(4 + 1 i) (4 - i)}$

Step 2.2.2.1.4.2

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

$\frac{24 - 6 i - 8 i + 2 (i^{1} i)}{(4 + 1 i) (4 - i)}$

Step 2.2.2.1.4.3

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

$\frac{24 - 6 i - 8 i + 2 (i^{1} i^{1})}{(4 + 1 i) (4 - i)}$

Step 2.2.2.1.4.4

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

$\frac{24 - 6 i - 8 i + 2 i^{1 + 1}}{(4 + 1 i) (4 - i)}$

Step 2.2.2.1.4.5

Add $1$ and $1$ .

$\frac{24 - 6 i - 8 i + 2 i^{2}}{(4 + 1 i) (4 - i)}$

Step 2.2.2.1.5

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

$\frac{24 - 6 i - 8 i + 2 \cdot - 1}{(4 + 1 i) (4 - i)}$

Step 2.2.2.1.6

Multiply $2$ by $- 1$ .

$\frac{24 - 6 i - 8 i - 2}{(4 + 1 i) (4 - i)}$

Step 2.2.2.2

Subtract $2$ from $24$ .

$\frac{22 - 6 i - 8 i}{(4 + 1 i) (4 - i)}$

Step 2.2.2.3

Subtract $8 i$ from $- 6 i$ .

$\frac{22 - 14 i}{(4 + 1 i) (4 - i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

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

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Step 2.3.1.1

Apply the distributive property.

$\frac{22 - 14 i}{4 (4 - i) + 1 i (4 - i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{22 - 14 i}{4 \cdot 4 + 4 (- i) + 1 i (4 - i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{22 - 14 i}{4 \cdot 4 + 4 (- i) + 1 i \cdot 4 + 1 i (- i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $4$ by $4$ .

$\frac{22 - 14 i}{16 + 4 (- i) + 1 i \cdot 4 + 1 i (- i)}$

Step 2.3.2.2

Multiply $- 1$ by $4$ .

$\frac{22 - 14 i}{16 - 4 i + 1 i \cdot 4 + 1 i (- i)}$

Step 2.3.2.3

Multiply $4$ by $1$ .

$\frac{22 - 14 i}{16 - 4 i + 4 i + 1 i (- i)}$

Step 2.3.2.4

Multiply $- 1$ by $1$ .

$\frac{22 - 14 i}{16 - 4 i + 4 i - i i}$

Step 2.3.2.5

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

$\frac{22 - 14 i}{16 - 4 i + 4 i - (i^{1} i)}$

Step 2.3.2.6

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

$\frac{22 - 14 i}{16 - 4 i + 4 i - (i^{1} i^{1})}$

Step 2.3.2.7

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

$\frac{22 - 14 i}{16 - 4 i + 4 i - i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{22 - 14 i}{16 - 4 i + 4 i - i^{2}}$

Step 2.3.2.9

Add $- 4 i$ and $4 i$ .

$\frac{22 - 14 i}{16 + 0 - i^{2}}$

Step 2.3.2.10

Add $16$ and $0$ .

$\frac{22 - 14 i}{16 - i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{22 - 14 i}{16 - - 1}$

Step 2.3.3.2

Multiply $- 1$ by $- 1$ .

$\frac{22 - 14 i}{16 + 1}$

Step 2.3.4

Add $16$ and $1$ .

$\frac{22 - 14 i}{17}$

Step 3

Split the fraction $\frac{22 - 14 i}{17}$ into two fractions.

$\frac{22}{17} + \frac{- 14 i}{17}$

Step 4

Move the negative in front of the fraction.

$\frac{22}{17} - \frac{14 i}{17}$