Precalculus Examples

$\frac{\sqrt{7} + 4 i}{5 - 3 i}$

Step 1

Multiply the numerator and denominator of $\frac{\sqrt{7} + 4 i}{5 - 3 i}$ by the conjugate of $5 - 3 i$ to make the denominator real.

$\frac{\sqrt{7} + 4 i}{5 - 3 i} \cdot \frac{5 + 3 i}{5 + 3 i}$

Step 2

Multiply.

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Step 2.1

Combine.

$\frac{(\sqrt{7} + 4 i) (5 + 3 i)}{(5 - 3 i) (5 + 3 i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand $(\sqrt{7} + 4 i) (5 + 3 i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

$\frac{\sqrt{7} (5 + 3 i) + 4 i (5 + 3 i)}{(5 - 3 i) (5 + 3 i)}$

Step 2.2.1.2

Apply the distributive property.

$\frac{\sqrt{7} \cdot 5 + \sqrt{7} (3 i) + 4 i (5 + 3 i)}{(5 - 3 i) (5 + 3 i)}$

Step 2.2.1.3

Apply the distributive property.

$\frac{\sqrt{7} \cdot 5 + \sqrt{7} (3 i) + 4 i \cdot 5 + 4 i (3 i)}{(5 - 3 i) (5 + 3 i)}$

Step 2.2.2

Simplify each term.

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Step 2.2.2.1

Move $5$ to the left of $\sqrt{7}$ .

$\frac{5 \cdot \sqrt{7} + \sqrt{7} (3 i) + 4 i \cdot 5 + 4 i (3 i)}{(5 - 3 i) (5 + 3 i)}$

Step 2.2.2.2

Move $3$ to the left of $\sqrt{7}$ .

$\frac{5 \sqrt{7} + 3 \cdot \sqrt{7} i + 4 i \cdot 5 + 4 i (3 i)}{(5 - 3 i) (5 + 3 i)}$

Step 2.2.2.3

Multiply $5$ by $4$ .

$\frac{5 \sqrt{7} + 3 \sqrt{7} i + 20 i + 4 i (3 i)}{(5 - 3 i) (5 + 3 i)}$

Step 2.2.2.4

Multiply $4 i (3 i)$ .

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Step 2.2.2.4.1

Multiply $3$ by $4$ .

$\frac{5 \sqrt{7} + 3 \sqrt{7} i + 20 i + 12 i i}{(5 - 3 i) (5 + 3 i)}$

Step 2.2.2.4.2

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

$\frac{5 \sqrt{7} + 3 \sqrt{7} i + 20 i + 12 (i^{1} i)}{(5 - 3 i) (5 + 3 i)}$

Step 2.2.2.4.3

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

$\frac{5 \sqrt{7} + 3 \sqrt{7} i + 20 i + 12 (i^{1} i^{1})}{(5 - 3 i) (5 + 3 i)}$

Step 2.2.2.4.4

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

$\frac{5 \sqrt{7} + 3 \sqrt{7} i + 20 i + 12 i^{1 + 1}}{(5 - 3 i) (5 + 3 i)}$

Step 2.2.2.4.5

Add $1$ and $1$ .

$\frac{5 \sqrt{7} + 3 \sqrt{7} i + 20 i + 12 i^{2}}{(5 - 3 i) (5 + 3 i)}$

Step 2.2.2.5

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

$\frac{5 \sqrt{7} + 3 \sqrt{7} i + 20 i + 12 \cdot - 1}{(5 - 3 i) (5 + 3 i)}$

Step 2.2.2.6

Multiply $12$ by $- 1$ .

$\frac{5 \sqrt{7} + 3 \sqrt{7} i + 20 i - 12}{(5 - 3 i) (5 + 3 i)}$

Step 2.2.3

Reorder $5 \sqrt{7} + 3 \sqrt{7} i + 20 i$ and $- 12$ .

$\frac{- 12 + 5 \sqrt{7} + 3 \sqrt{7} i + 20 i}{(5 - 3 i) (5 + 3 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand $(5 - 3 i) (5 + 3 i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

$\frac{- 12 + 5 \sqrt{7} + 3 \sqrt{7} i + 20 i}{5 (5 + 3 i) - 3 i (5 + 3 i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{- 12 + 5 \sqrt{7} + 3 \sqrt{7} i + 20 i}{5 \cdot 5 + 5 (3 i) - 3 i (5 + 3 i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{- 12 + 5 \sqrt{7} + 3 \sqrt{7} i + 20 i}{5 \cdot 5 + 5 (3 i) - 3 i \cdot 5 - 3 i (3 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $5$ by $5$ .

$\frac{- 12 + 5 \sqrt{7} + 3 \sqrt{7} i + 20 i}{25 + 5 (3 i) - 3 i \cdot 5 - 3 i (3 i)}$

Step 2.3.2.2

Multiply $3$ by $5$ .

$\frac{- 12 + 5 \sqrt{7} + 3 \sqrt{7} i + 20 i}{25 + 15 i - 3 i \cdot 5 - 3 i (3 i)}$

Step 2.3.2.3

Multiply $5$ by $- 3$ .

$\frac{- 12 + 5 \sqrt{7} + 3 \sqrt{7} i + 20 i}{25 + 15 i - 15 i - 3 i (3 i)}$

Step 2.3.2.4

Multiply $3$ by $- 3$ .

$\frac{- 12 + 5 \sqrt{7} + 3 \sqrt{7} i + 20 i}{25 + 15 i - 15 i - 9 i i}$

Step 2.3.2.5

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

$\frac{- 12 + 5 \sqrt{7} + 3 \sqrt{7} i + 20 i}{25 + 15 i - 15 i - 9 (i^{1} i)}$

Step 2.3.2.6

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

$\frac{- 12 + 5 \sqrt{7} + 3 \sqrt{7} i + 20 i}{25 + 15 i - 15 i - 9 (i^{1} i^{1})}$

Step 2.3.2.7

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

$\frac{- 12 + 5 \sqrt{7} + 3 \sqrt{7} i + 20 i}{25 + 15 i - 15 i - 9 i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{- 12 + 5 \sqrt{7} + 3 \sqrt{7} i + 20 i}{25 + 15 i - 15 i - 9 i^{2}}$

Step 2.3.2.9

Subtract $15 i$ from $15 i$ .

$\frac{- 12 + 5 \sqrt{7} + 3 \sqrt{7} i + 20 i}{25 + 0 - 9 i^{2}}$

Step 2.3.2.10

Add $25$ and $0$ .

$\frac{- 12 + 5 \sqrt{7} + 3 \sqrt{7} i + 20 i}{25 - 9 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{- 12 + 5 \sqrt{7} + 3 \sqrt{7} i + 20 i}{25 - 9 \cdot - 1}$

Step 2.3.3.2

Multiply $- 9$ by $- 1$ .

$\frac{- 12 + 5 \sqrt{7} + 3 \sqrt{7} i + 20 i}{25 + 9}$

Step 2.3.4

Add $25$ and $9$ .

$\frac{- 12 + 5 \sqrt{7} + 3 \sqrt{7} i + 20 i}{34}$

Step 3

Rewrite $- 12$ as $- 1 (12)$ .

$\frac{- 1 (12) + 5 \sqrt{7} + 3 \sqrt{7} i + 20 i}{34}$

Step 4

Factor $- 1$ out of $5 \sqrt{7}$ .

$\frac{- 1 (12) - (- 5 \sqrt{7}) + 3 \sqrt{7} i + 20 i}{34}$

Step 5

Factor $- 1$ out of $- 1 (12) - (- 5 \sqrt{7})$ .

$\frac{- 1 (12 - 5 \sqrt{7}) + 3 \sqrt{7} i + 20 i}{34}$

Step 6

Factor $- 1$ out of $3 \sqrt{7} i$ .

$\frac{- 1 (12 - 5 \sqrt{7}) - (- 3 \sqrt{7} i) + 20 i}{34}$

Step 7

Factor $- 1$ out of $- 1 (12 - 5 \sqrt{7}) - (- 3 \sqrt{7} i)$ .

$\frac{- 1 (12 - 5 \sqrt{7} - 3 \sqrt{7} i) + 20 i}{34}$

Step 8

Factor $- 1$ out of $20 i$ .

$\frac{- 1 (12 - 5 \sqrt{7} - 3 \sqrt{7} i) - (- 20 i)}{34}$

Step 9

Factor $- 1$ out of $- 1 (12 - 5 \sqrt{7} - 3 \sqrt{7} i) - (- 20 i)$ .

$\frac{- 1 (12 - 5 \sqrt{7} - 3 \sqrt{7} i - 20 i)}{34}$

Step 10

Move the negative in front of the fraction.

$- \frac{12 - 5 \sqrt{7} - 3 \sqrt{7} i - 20 i}{34}$

Step 11

Split the fraction $\frac{12 - 5 \sqrt{7} - 3 \sqrt{7} i - 20 i}{34}$ into two fractions.

$- (\frac{12 - 5 \sqrt{7} - 3 \sqrt{7} i}{34} + \frac{- 20 i}{34})$

Step 12

Split the fraction $\frac{12 - 5 \sqrt{7} - 3 \sqrt{7} i}{34}$ into two fractions.

$- (\frac{12 - 5 \sqrt{7}}{34} + \frac{- 3 \sqrt{7} i}{34} + \frac{- 20 i}{34})$

Step 13

Split the fraction $\frac{12 - 5 \sqrt{7}}{34}$ into two fractions.

$- (\frac{12}{34} + \frac{- 5 \sqrt{7}}{34} + \frac{- 3 \sqrt{7} i}{34} + \frac{- 20 i}{34})$

Step 14

Cancel the common factor of $12$ and $34$ .

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Step 14.1

Factor $2$ out of $12$ .

$- (\frac{2 (6)}{34} + \frac{- 5 \sqrt{7}}{34} + \frac{- 3 \sqrt{7} i}{34} + \frac{- 20 i}{34})$

Step 14.2

Cancel the common factors.

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Step 14.2.1

Factor $2$ out of $34$ .

$- (\frac{2 \cdot 6}{2 \cdot 17} + \frac{- 5 \sqrt{7}}{34} + \frac{- 3 \sqrt{7} i}{34} + \frac{- 20 i}{34})$

Step 14.2.2

Cancel the common factor.

$- (\frac{2 \cdot 6}{2 \cdot 17} + \frac{- 5 \sqrt{7}}{34} + \frac{- 3 \sqrt{7} i}{34} + \frac{- 20 i}{34})$

Step 14.2.3

Rewrite the expression.

$- (\frac{6}{17} + \frac{- 5 \sqrt{7}}{34} + \frac{- 3 \sqrt{7} i}{34} + \frac{- 20 i}{34})$

Step 15

Move the negative in front of the fraction.

$- (\frac{6}{17} - \frac{5 \sqrt{7}}{34} + \frac{- 3 \sqrt{7} i}{34} + \frac{- 20 i}{34})$

Step 16

Move the negative in front of the fraction.

$- (\frac{6}{17} - \frac{5 \sqrt{7}}{34} - \frac{3 \sqrt{7} i}{34} + \frac{- 20 i}{34})$

Step 17

Cancel the common factor of $- 20$ and $34$ .

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Step 17.1

Factor $2$ out of $- 20 i$ .

$- (\frac{6}{17} - \frac{5 \sqrt{7}}{34} - \frac{3 \sqrt{7} i}{34} + \frac{2 (- 10 i)}{34})$

Step 17.2

Cancel the common factors.

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Step 17.2.1

Factor $2$ out of $34$ .

$- (\frac{6}{17} - \frac{5 \sqrt{7}}{34} - \frac{3 \sqrt{7} i}{34} + \frac{2 (- 10 i)}{2 (17)})$

Step 17.2.2

Cancel the common factor.

$- (\frac{6}{17} - \frac{5 \sqrt{7}}{34} - \frac{3 \sqrt{7} i}{34} + \frac{2 (- 10 i)}{2 \cdot 17})$

Step 17.2.3

Rewrite the expression.

$- (\frac{6}{17} - \frac{5 \sqrt{7}}{34} - \frac{3 \sqrt{7} i}{34} + \frac{- 10 i}{17})$

Step 18

Move the negative in front of the fraction.

$- (\frac{6}{17} - \frac{5 \sqrt{7}}{34} - \frac{3 \sqrt{7} i}{34} - \frac{10 i}{17})$