Precalculus Examples

$\frac{7}{4 - 14 i}$

Step 1

Multiply the numerator and denominator of $\frac{7}{4 - 14 i}$ by the conjugate of $4 - 14 i$ to make the denominator real.

$\frac{7}{4 - 14 i} \cdot \frac{4 + 14 i}{4 + 14 i}$

Step 2

Multiply.

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Step 2.1

Combine.

$\frac{7 (4 + 14 i)}{(4 - 14 i) (4 + 14 i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the distributive property.

$\frac{7 \cdot 4 + 7 (14 i)}{(4 - 14 i) (4 + 14 i)}$

Step 2.2.2

Multiply $7$ by $4$ .

$\frac{28 + 7 (14 i)}{(4 - 14 i) (4 + 14 i)}$

Step 2.2.3

Multiply $14$ by $7$ .

$\frac{28 + 98 i}{(4 - 14 i) (4 + 14 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand $(4 - 14 i) (4 + 14 i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

$\frac{28 + 98 i}{4 (4 + 14 i) - 14 i (4 + 14 i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{28 + 98 i}{4 \cdot 4 + 4 (14 i) - 14 i (4 + 14 i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{28 + 98 i}{4 \cdot 4 + 4 (14 i) - 14 i \cdot 4 - 14 i (14 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $4$ by $4$ .

$\frac{28 + 98 i}{16 + 4 (14 i) - 14 i \cdot 4 - 14 i (14 i)}$

Step 2.3.2.2

Multiply $14$ by $4$ .

$\frac{28 + 98 i}{16 + 56 i - 14 i \cdot 4 - 14 i (14 i)}$

Step 2.3.2.3

Multiply $4$ by $- 14$ .

$\frac{28 + 98 i}{16 + 56 i - 56 i - 14 i (14 i)}$

Step 2.3.2.4

Multiply $14$ by $- 14$ .

$\frac{28 + 98 i}{16 + 56 i - 56 i - 196 i i}$

Step 2.3.2.5

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

$\frac{28 + 98 i}{16 + 56 i - 56 i - 196 (i^{1} i)}$

Step 2.3.2.6

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

$\frac{28 + 98 i}{16 + 56 i - 56 i - 196 (i^{1} i^{1})}$

Step 2.3.2.7

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

$\frac{28 + 98 i}{16 + 56 i - 56 i - 196 i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{28 + 98 i}{16 + 56 i - 56 i - 196 i^{2}}$

Step 2.3.2.9

Subtract $56 i$ from $56 i$ .

$\frac{28 + 98 i}{16 + 0 - 196 i^{2}}$

Step 2.3.2.10

Add $16$ and $0$ .

$\frac{28 + 98 i}{16 - 196 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{28 + 98 i}{16 - 196 \cdot - 1}$

Step 2.3.3.2

Multiply $- 196$ by $- 1$ .

$\frac{28 + 98 i}{16 + 196}$

Step 2.3.4

Add $16$ and $196$ .

$\frac{28 + 98 i}{212}$

Step 3

Cancel the common factor of $28 + 98 i$ and $212$ .

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Step 3.1

Factor $2$ out of $28$ .

$\frac{2 (14) + 98 i}{212}$

Step 3.2

Factor $2$ out of $98 i$ .

$\frac{2 (14) + 2 (49 i)}{212}$

Step 3.3

Factor $2$ out of $2 (14) + 2 (49 i)$ .

$\frac{2 (14 + 49 i)}{212}$

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor $2$ out of $212$ .

$\frac{2 (14 + 49 i)}{2 \cdot 106}$

Step 3.4.2

Cancel the common factor.

$\frac{2 (14 + 49 i)}{2 \cdot 106}$

Step 3.4.3

Rewrite the expression.

$\frac{14 + 49 i}{106}$

Step 4

Split the fraction $\frac{14 + 49 i}{106}$ into two fractions.

$\frac{14}{106} + \frac{49 i}{106}$

Step 5

Cancel the common factor of $14$ and $106$ .

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Step 5.1

Factor $2$ out of $14$ .

$\frac{2 (7)}{106} + \frac{49 i}{106}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $2$ out of $106$ .

$\frac{2 \cdot 7}{2 \cdot 53} + \frac{49 i}{106}$

Step 5.2.2

Cancel the common factor.

$\frac{2 \cdot 7}{2 \cdot 53} + \frac{49 i}{106}$

Step 5.2.3

Rewrite the expression.

$\frac{7}{53} + \frac{49 i}{106}$