Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Multiply.

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Step 1.2.1

Combine.

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

Step 1.2.2

Multiply $1 - i$ by $1$ .

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

Step 1.2.3

Simplify the denominator.

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Step 1.2.3.1

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

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Step 1.2.3.1.1

Apply the distributive property.

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

Step 1.2.3.1.2

Apply the distributive property.

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

Step 1.2.3.1.3

Apply the distributive property.

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

Step 1.2.3.2

Simplify.

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Step 1.2.3.2.1

Multiply $1$ by $1$ .

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

Step 1.2.3.2.2

Multiply $- 1$ by $1$ .

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

Step 1.2.3.2.3

Multiply $1$ by $1$ .

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

Step 1.2.3.2.4

Multiply $- 1$ by $1$ .

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

Step 1.2.3.2.5

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

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

Step 1.2.3.2.6

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

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

Step 1.2.3.2.7

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

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

Step 1.2.3.2.8

Add $1$ and $1$ .

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

Step 1.2.3.2.9

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

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

Step 1.2.3.2.10

Add $1$ and $0$ .

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

Step 1.2.3.3

Simplify each term.

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Step 1.2.3.3.1

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

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

Step 1.2.3.3.2

Multiply $- 1$ by $- 1$ .

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

Step 1.2.3.4

Add $1$ and $1$ .

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

Step 1.3

Split the fraction $\frac{1 - i}{2}$ into two fractions.

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

Step 1.4

Move the negative in front of the fraction.

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

Step 1.5

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

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

Step 1.6

Multiply.

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Step 1.6.1

Combine.

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

Step 1.6.2

Multiply $1 + i$ by $1$ .

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

Step 1.6.3

Simplify the denominator.

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Step 1.6.3.1

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

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Step 1.6.3.1.1

Apply the distributive property.

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

Step 1.6.3.1.2

Apply the distributive property.

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

Step 1.6.3.1.3

Apply the distributive property.

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

Step 1.6.3.2

Simplify.

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Step 1.6.3.2.1

Multiply $1$ by $1$ .

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

Step 1.6.3.2.2

Multiply $- 1$ by $1$ .

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

Step 1.6.3.2.3

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

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

Step 1.6.3.2.4

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

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

Step 1.6.3.2.5

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

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

Step 1.6.3.2.6

Add $1$ and $1$ .

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

Step 1.6.3.2.7

Subtract $i$ from $1 i$ .

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

Step 1.6.3.2.8

Add $1$ and $0$ .

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

Step 1.6.3.3

Simplify each term.

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Step 1.6.3.3.1

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

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

Step 1.6.3.3.2

Multiply $- 1$ by $- 1$ .

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

Step 1.6.3.4

Add $1$ and $1$ .

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

Step 1.7

Split the fraction $\frac{1 + i}{2}$ into two fractions.

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

Step 1.8

Apply the distributive property.

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

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

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

Step 2.2

Simplify by subtracting numbers.

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Step 2.2.1

Subtract $1$ from $1$ .

$\frac{0 - i - i}{2}$

Step 2.2.2

Subtract $i$ from $0$ .

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

Step 2.3

Subtract $i$ from $- i$ .

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

Cancel the common factors.

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Step 2.4.2.1

Factor $2$ out of $2$ .

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

Step 2.4.2.2

Cancel the common factor.

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

Step 2.4.2.3

Rewrite the expression.

$\frac{- i}{1}$

Step 2.4.2.4

Divide $- i$ by $1$ .

$- i$