Precalculus Examples

$\frac{29}{1 - i}$

Step 1

Multiply the numerator and denominator of $\frac{29}{1 - i}$ by the conjugate of $1 - i$ to make the denominator real.

$\frac{29}{1 - i} \cdot \frac{1 + i}{1 + i}$

Step 2

Multiply.

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Step 2.1

Combine.

$\frac{29 (1 + i)}{(1 - i) (1 + i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the distributive property.

$\frac{29 \cdot 1 + 29 i}{(1 - i) (1 + i)}$

Step 2.2.2

Multiply $29$ by $1$ .

$\frac{29 + 29 i}{(1 - i) (1 + i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

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

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Step 2.3.1.1

Apply the distributive property.

$\frac{29 + 29 i}{1 (1 + i) - i (1 + i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{29 + 29 i}{1 \cdot 1 + 1 i - i (1 + i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{29 + 29 i}{1 \cdot 1 + 1 i - i \cdot 1 - i i}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $1$ by $1$ .

$\frac{29 + 29 i}{1 + 1 i - i \cdot 1 - i i}$

Step 2.3.2.2

Multiply $- 1$ by $1$ .

$\frac{29 + 29 i}{1 + 1 i - i - i i}$

Step 2.3.2.3

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

$\frac{29 + 29 i}{1 + 1 i - i - (i^{1} i)}$

Step 2.3.2.4

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

$\frac{29 + 29 i}{1 + 1 i - i - (i^{1} i^{1})}$

Step 2.3.2.5

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

$\frac{29 + 29 i}{1 + 1 i - i - i^{1 + 1}}$

Step 2.3.2.6

Add $1$ and $1$ .

$\frac{29 + 29 i}{1 + 1 i - i - i^{2}}$

Step 2.3.2.7

Subtract $i$ from $1 i$ .

$\frac{29 + 29 i}{1 + 0 - i^{2}}$

Step 2.3.2.8

Add $1$ and $0$ .

$\frac{29 + 29 i}{1 - i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{29 + 29 i}{1 - - 1}$

Step 2.3.3.2

Multiply $- 1$ by $- 1$ .

$\frac{29 + 29 i}{1 + 1}$

Step 2.3.4

Add $1$ and $1$ .

$\frac{29 + 29 i}{2}$

Step 3

Split the fraction $\frac{29 + 29 i}{2}$ into two fractions.

$\frac{29}{2} + \frac{29 i}{2}$