Calculus Examples

$\frac{2 - 4 i}{5 + 3 i}$

Step 1

Multiply the numerator and denominator of $\frac{2 - 4 i}{5 + 3 i}$ by the conjugate of $5 + 3 i$ to make the denominator real.

$\frac{2 - 4 i}{5 + 3 i} \cdot \frac{5 - 3 i}{5 - 3 i}$

Step 2

Multiply.

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Step 2.1

Combine.

$\frac{(2 - 4 i) (5 - 3 i)}{(5 + 3 i) (5 - 3 i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand $(2 - 4 i) (5 - 3 i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

$\frac{2 (5 - 3 i) - 4 i (5 - 3 i)}{(5 + 3 i) (5 - 3 i)}$

Step 2.2.1.2

Apply the distributive property.

$\frac{2 \cdot 5 + 2 (- 3 i) - 4 i (5 - 3 i)}{(5 + 3 i) (5 - 3 i)}$

Step 2.2.1.3

Apply the distributive property.

$\frac{2 \cdot 5 + 2 (- 3 i) - 4 i \cdot 5 - 4 i (- 3 i)}{(5 + 3 i) (5 - 3 i)}$

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 $2$ by $5$ .

$\frac{10 + 2 (- 3 i) - 4 i \cdot 5 - 4 i (- 3 i)}{(5 + 3 i) (5 - 3 i)}$

Step 2.2.2.1.2

Multiply $- 3$ by $2$ .

$\frac{10 - 6 i - 4 i \cdot 5 - 4 i (- 3 i)}{(5 + 3 i) (5 - 3 i)}$

Step 2.2.2.1.3

Multiply $5$ by $- 4$ .

$\frac{10 - 6 i - 20 i - 4 i (- 3 i)}{(5 + 3 i) (5 - 3 i)}$

Step 2.2.2.1.4

Multiply $- 4 i (- 3 i)$ .

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Step 2.2.2.1.4.1

Multiply $- 3$ by $- 4$ .

$\frac{10 - 6 i - 20 i + 12 i i}{(5 + 3 i) (5 - 3 i)}$

Step 2.2.2.1.4.2

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

$\frac{10 - 6 i - 20 i + 12 (i^{1} i)}{(5 + 3 i) (5 - 3 i)}$

Step 2.2.2.1.4.3

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

$\frac{10 - 6 i - 20 i + 12 (i^{1} i^{1})}{(5 + 3 i) (5 - 3 i)}$

Step 2.2.2.1.4.4

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

$\frac{10 - 6 i - 20 i + 12 i^{1 + 1}}{(5 + 3 i) (5 - 3 i)}$

Step 2.2.2.1.4.5

Add $1$ and $1$ .

$\frac{10 - 6 i - 20 i + 12 i^{2}}{(5 + 3 i) (5 - 3 i)}$

Step 2.2.2.1.5

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

$\frac{10 - 6 i - 20 i + 12 \cdot - 1}{(5 + 3 i) (5 - 3 i)}$

Step 2.2.2.1.6

Multiply $12$ by $- 1$ .

$\frac{10 - 6 i - 20 i - 12}{(5 + 3 i) (5 - 3 i)}$

Step 2.2.2.2

Subtract $12$ from $10$ .

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

Step 2.2.2.3

Subtract $20 i$ from $- 6 i$ .

$\frac{- 2 - 26 i}{(5 + 3 i) (5 - 3 i)}$

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.

$\frac{- 2 - 26 i}{5 (5 - 3 i) + 3 i (5 - 3 i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{- 2 - 26 i}{5 \cdot 5 + 5 (- 3 i) + 3 i (5 - 3 i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{- 2 - 26 i}{5 \cdot 5 + 5 (- 3 i) + 3 i \cdot 5 + 3 i (- 3 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $5$ by $5$ .

$\frac{- 2 - 26 i}{25 + 5 (- 3 i) + 3 i \cdot 5 + 3 i (- 3 i)}$

Step 2.3.2.2

Multiply $- 3$ by $5$ .

$\frac{- 2 - 26 i}{25 - 15 i + 3 i \cdot 5 + 3 i (- 3 i)}$

Step 2.3.2.3

Multiply $5$ by $3$ .

$\frac{- 2 - 26 i}{25 - 15 i + 15 i + 3 i (- 3 i)}$

Step 2.3.2.4

Multiply $- 3$ by $3$ .

$\frac{- 2 - 26 i}{25 - 15 i + 15 i - 9 i i}$

Step 2.3.2.5

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

$\frac{- 2 - 26 i}{25 - 15 i + 15 i - 9 (i^{1} i)}$

Step 2.3.2.6

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

$\frac{- 2 - 26 i}{25 - 15 i + 15 i - 9 (i^{1} i^{1})}$

Step 2.3.2.7

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

$\frac{- 2 - 26 i}{25 - 15 i + 15 i - 9 i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{- 2 - 26 i}{25 - 15 i + 15 i - 9 i^{2}}$

Step 2.3.2.9

Add $- 15 i$ and $15 i$ .

$\frac{- 2 - 26 i}{25 + 0 - 9 i^{2}}$

Step 2.3.2.10

Add $25$ and $0$ .

$\frac{- 2 - 26 i}{25 - 9 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{- 2 - 26 i}{25 - 9 \cdot - 1}$

Step 2.3.3.2

Multiply $- 9$ by $- 1$ .

$\frac{- 2 - 26 i}{25 + 9}$

Step 2.3.4

Add $25$ and $9$ .

$\frac{- 2 - 26 i}{34}$

Step 3

Cancel the common factor of $- 2 - 26 i$ and $34$ .

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Step 3.1

Factor $2$ out of $- 2$ .

$\frac{2 (- 1) - 26 i}{34}$

Step 3.2

Factor $2$ out of $- 26 i$ .

$\frac{2 (- 1) + 2 (- 13 i)}{34}$

Step 3.3

Factor $2$ out of $2 (- 1) + 2 (- 13 i)$ .

$\frac{2 (- 1 - 13 i)}{34}$

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor $2$ out of $34$ .

$\frac{2 (- 1 - 13 i)}{2 \cdot 17}$

Step 3.4.2

Cancel the common factor.

$\frac{2 (- 1 - 13 i)}{2 \cdot 17}$

Step 3.4.3

Rewrite the expression.

$\frac{- 1 - 13 i}{17}$

Step 4

Split the fraction $\frac{- 1 - 13 i}{17}$ into two fractions.

$\frac{- 1}{17} + \frac{- 13 i}{17}$

Step 5

Simplify each term.

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Step 5.1

Move the negative in front of the fraction.

$- \frac{1}{17} + \frac{- 13 i}{17}$

Step 5.2

Move the negative in front of the fraction.

$- \frac{1}{17} - \frac{13 i}{17}$