Finite Math Examples

$\frac{4}{5 + 6 i}$

Step 1

Multiply the numerator and denominator of $\frac{4}{5 + 6 i}$ by the conjugate of $5 + 6 i$ to make the denominator real.

$\frac{4}{5 + 6 i} \cdot \frac{5 - 6 i}{5 - 6 i}$

Step 2

Multiply.

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Step 2.1

Combine.

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Multiply $4$ by $5$ .

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

Step 2.2.3

Multiply $- 6$ by $4$ .

$\frac{20 - 24 i}{(5 + 6 i) (5 - 6 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand $(5 + 6 i) (5 - 6 i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

$\frac{20 - 24 i}{5 (5 - 6 i) + 6 i (5 - 6 i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{20 - 24 i}{5 \cdot 5 + 5 (- 6 i) + 6 i (5 - 6 i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{20 - 24 i}{5 \cdot 5 + 5 (- 6 i) + 6 i \cdot 5 + 6 i (- 6 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $5$ by $5$ .

$\frac{20 - 24 i}{25 + 5 (- 6 i) + 6 i \cdot 5 + 6 i (- 6 i)}$

Step 2.3.2.2

Multiply $- 6$ by $5$ .

$\frac{20 - 24 i}{25 - 30 i + 6 i \cdot 5 + 6 i (- 6 i)}$

Step 2.3.2.3

Multiply $5$ by $6$ .

$\frac{20 - 24 i}{25 - 30 i + 30 i + 6 i (- 6 i)}$

Step 2.3.2.4

Multiply $- 6$ by $6$ .

$\frac{20 - 24 i}{25 - 30 i + 30 i - 36 i i}$

Step 2.3.2.5

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

$\frac{20 - 24 i}{25 - 30 i + 30 i - 36 (i^{1} i)}$

Step 2.3.2.6

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

$\frac{20 - 24 i}{25 - 30 i + 30 i - 36 (i^{1} i^{1})}$

Step 2.3.2.7

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

$\frac{20 - 24 i}{25 - 30 i + 30 i - 36 i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{20 - 24 i}{25 - 30 i + 30 i - 36 i^{2}}$

Step 2.3.2.9

Add $- 30 i$ and $30 i$ .

$\frac{20 - 24 i}{25 + 0 - 36 i^{2}}$

Step 2.3.2.10

Add $25$ and $0$ .

$\frac{20 - 24 i}{25 - 36 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{20 - 24 i}{25 - 36 \cdot - 1}$

Step 2.3.3.2

Multiply $- 36$ by $- 1$ .

$\frac{20 - 24 i}{25 + 36}$

Step 2.3.4

Add $25$ and $36$ .

$\frac{20 - 24 i}{61}$

Step 3

Split the fraction $\frac{20 - 24 i}{61}$ into two fractions.

$\frac{20}{61} + \frac{- 24 i}{61}$

Step 4

Move the negative in front of the fraction.

$\frac{20}{61} - \frac{24 i}{61}$