Algebra Examples

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

Step 1

Multiply the numerator and denominator of $\frac{3 - i}{1 + 1 i}$ by the conjugate of $1 + 1 i$ to make the denominator real.

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

Step 2

Multiply.

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Step 2.1

Combine.

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand $(3 - i) (1 - i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

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

Step 2.2.1.2

Apply the distributive property.

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

Step 2.2.1.3

Apply the distributive property.

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

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 $3$ by $1$ .

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

Step 2.2.2.1.2

Multiply $- 1$ by $3$ .

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

Step 2.2.2.1.3

Multiply $- 1$ by $1$ .

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

Step 2.2.2.1.4

Multiply $- i (- i)$ .

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Step 2.2.2.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 2.2.2.1.4.2

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

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

Step 2.2.2.1.4.3

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

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

Step 2.2.2.1.4.4

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

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

Step 2.2.2.1.4.5

Add $1$ and $1$ .

$\frac{3 - 3 i - i + 1 i^{2}}{(1 + 1 i) (1 - i)}$

Step 2.2.2.1.4.6

Multiply $i^{2}$ by $1$ .

$\frac{3 - 3 i - i + i^{2}}{(1 + 1 i) (1 - i)}$

Step 2.2.2.1.5

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

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

Step 2.2.2.2

Subtract $1$ from $3$ .

$\frac{2 - 3 i - i}{(1 + 1 i) (1 - i)}$

Step 2.2.2.3

Subtract $i$ from $- 3 i$ .

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

Step 2.3

Simplify the denominator.

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Step 2.3.1

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

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Step 2.3.1.1

Apply the distributive property.

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

Step 2.3.1.2

Apply the distributive property.

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

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $1$ by $1$ .

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

Step 2.3.2.2

Multiply $- 1$ by $1$ .

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

Step 2.3.2.3

Multiply $1$ by $1$ .

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

Step 2.3.2.4

Multiply $- 1$ by $1$ .

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

Step 2.3.2.5

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

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

Step 2.3.2.6

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

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

Step 2.3.2.7

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

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

Step 2.3.2.8

Add $1$ and $1$ .

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

Step 2.3.2.9

Add $- 1 i$ and $1 i$ .

$\frac{2 - 4 i}{1 + 0 - i^{2}}$

Step 2.3.2.10

Add $1$ and $0$ .

$\frac{2 - 4 i}{1 - i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{2 - 4 i}{1 - - 1}$

Step 2.3.3.2

Multiply $- 1$ by $- 1$ .

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

Step 2.3.4

Add $1$ and $1$ .

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

Step 3

Cancel the common factor of $2 - 4 i$ and $2$ .

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Step 3.1

Factor $2$ out of $2$ .

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

Step 3.2

Factor $2$ out of $- 4 i$ .

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

Step 3.3

Factor $2$ out of $2 (1) + 2 (- 2 i)$ .

$\frac{2 (1 - 2 i)}{2}$

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor $2$ out of $2$ .

$\frac{2 (1 - 2 i)}{2 (1)}$

Step 3.4.2

Cancel the common factor.

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

Step 3.4.3

Rewrite the expression.

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

Step 3.4.4

Divide $1 - 2 i$ by $1$ .

$1 - 2 i$