Finite Math Examples

$\frac{6 - 5 i}{4 + 2 i}$

Step 1

Multiply the numerator and denominator of $\frac{6 - 5 i}{4 + 2 i}$ by the conjugate of $4 + 2 i$ to make the denominator real.

$\frac{6 - 5 i}{4 + 2 i} \cdot \frac{4 - 2 i}{4 - 2 i}$

Step 2

Multiply.

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Step 2.1

Combine.

$\frac{(6 - 5 i) (4 - 2 i)}{(4 + 2 i) (4 - 2 i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand $(6 - 5 i) (4 - 2 i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

$\frac{6 (4 - 2 i) - 5 i (4 - 2 i)}{(4 + 2 i) (4 - 2 i)}$

Step 2.2.1.2

Apply the distributive property.

$\frac{6 \cdot 4 + 6 (- 2 i) - 5 i (4 - 2 i)}{(4 + 2 i) (4 - 2 i)}$

Step 2.2.1.3

Apply the distributive property.

$\frac{6 \cdot 4 + 6 (- 2 i) - 5 i \cdot 4 - 5 i (- 2 i)}{(4 + 2 i) (4 - 2 i)}$

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 $6$ by $4$ .

$\frac{24 + 6 (- 2 i) - 5 i \cdot 4 - 5 i (- 2 i)}{(4 + 2 i) (4 - 2 i)}$

Step 2.2.2.1.2

Multiply $- 2$ by $6$ .

$\frac{24 - 12 i - 5 i \cdot 4 - 5 i (- 2 i)}{(4 + 2 i) (4 - 2 i)}$

Step 2.2.2.1.3

Multiply $4$ by $- 5$ .

$\frac{24 - 12 i - 20 i - 5 i (- 2 i)}{(4 + 2 i) (4 - 2 i)}$

Step 2.2.2.1.4

Multiply $- 5 i (- 2 i)$ .

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Step 2.2.2.1.4.1

Multiply $- 2$ by $- 5$ .

$\frac{24 - 12 i - 20 i + 10 i i}{(4 + 2 i) (4 - 2 i)}$

Step 2.2.2.1.4.2

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

$\frac{24 - 12 i - 20 i + 10 (i^{1} i)}{(4 + 2 i) (4 - 2 i)}$

Step 2.2.2.1.4.3

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

$\frac{24 - 12 i - 20 i + 10 (i^{1} i^{1})}{(4 + 2 i) (4 - 2 i)}$

Step 2.2.2.1.4.4

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

$\frac{24 - 12 i - 20 i + 10 i^{1 + 1}}{(4 + 2 i) (4 - 2 i)}$

Step 2.2.2.1.4.5

Add $1$ and $1$ .

$\frac{24 - 12 i - 20 i + 10 i^{2}}{(4 + 2 i) (4 - 2 i)}$

Step 2.2.2.1.5

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

$\frac{24 - 12 i - 20 i + 10 \cdot - 1}{(4 + 2 i) (4 - 2 i)}$

Step 2.2.2.1.6

Multiply $10$ by $- 1$ .

$\frac{24 - 12 i - 20 i - 10}{(4 + 2 i) (4 - 2 i)}$

Step 2.2.2.2

Subtract $10$ from $24$ .

$\frac{14 - 12 i - 20 i}{(4 + 2 i) (4 - 2 i)}$

Step 2.2.2.3

Subtract $20 i$ from $- 12 i$ .

$\frac{14 - 32 i}{(4 + 2 i) (4 - 2 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

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

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Step 2.3.1.1

Apply the distributive property.

$\frac{14 - 32 i}{4 (4 - 2 i) + 2 i (4 - 2 i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{14 - 32 i}{4 \cdot 4 + 4 (- 2 i) + 2 i (4 - 2 i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{14 - 32 i}{4 \cdot 4 + 4 (- 2 i) + 2 i \cdot 4 + 2 i (- 2 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $4$ by $4$ .

$\frac{14 - 32 i}{16 + 4 (- 2 i) + 2 i \cdot 4 + 2 i (- 2 i)}$

Step 2.3.2.2

Multiply $- 2$ by $4$ .

$\frac{14 - 32 i}{16 - 8 i + 2 i \cdot 4 + 2 i (- 2 i)}$

Step 2.3.2.3

Multiply $4$ by $2$ .

$\frac{14 - 32 i}{16 - 8 i + 8 i + 2 i (- 2 i)}$

Step 2.3.2.4

Multiply $- 2$ by $2$ .

$\frac{14 - 32 i}{16 - 8 i + 8 i - 4 i i}$

Step 2.3.2.5

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

$\frac{14 - 32 i}{16 - 8 i + 8 i - 4 (i^{1} i)}$

Step 2.3.2.6

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

$\frac{14 - 32 i}{16 - 8 i + 8 i - 4 (i^{1} i^{1})}$

Step 2.3.2.7

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

$\frac{14 - 32 i}{16 - 8 i + 8 i - 4 i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{14 - 32 i}{16 - 8 i + 8 i - 4 i^{2}}$

Step 2.3.2.9

Add $- 8 i$ and $8 i$ .

$\frac{14 - 32 i}{16 + 0 - 4 i^{2}}$

Step 2.3.2.10

Add $16$ and $0$ .

$\frac{14 - 32 i}{16 - 4 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{14 - 32 i}{16 - 4 \cdot - 1}$

Step 2.3.3.2

Multiply $- 4$ by $- 1$ .

$\frac{14 - 32 i}{16 + 4}$

Step 2.3.4

Add $16$ and $4$ .

$\frac{14 - 32 i}{20}$

Step 3

Cancel the common factor of $14 - 32 i$ and $20$ .

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Step 3.1

Factor $2$ out of $14$ .

$\frac{2 (7) - 32 i}{20}$

Step 3.2

Factor $2$ out of $- 32 i$ .

$\frac{2 (7) + 2 (- 16 i)}{20}$

Step 3.3

Factor $2$ out of $2 (7) + 2 (- 16 i)$ .

$\frac{2 (7 - 16 i)}{20}$

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor $2$ out of $20$ .

$\frac{2 (7 - 16 i)}{2 \cdot 10}$

Step 3.4.2

Cancel the common factor.

$\frac{2 (7 - 16 i)}{2 \cdot 10}$

Step 3.4.3

Rewrite the expression.

$\frac{7 - 16 i}{10}$

Step 4

Split the fraction $\frac{7 - 16 i}{10}$ into two fractions.

$\frac{7}{10} + \frac{- 16 i}{10}$

Step 5

Simplify each term.

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Step 5.1

Cancel the common factor of $- 16$ and $10$ .

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Step 5.1.1

Factor $2$ out of $- 16 i$ .

$\frac{7}{10} + \frac{2 (- 8 i)}{10}$

Step 5.1.2

Cancel the common factors.

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Step 5.1.2.1

Factor $2$ out of $10$ .

$\frac{7}{10} + \frac{2 (- 8 i)}{2 (5)}$

Step 5.1.2.2

Cancel the common factor.

$\frac{7}{10} + \frac{2 (- 8 i)}{2 \cdot 5}$

Step 5.1.2.3

Rewrite the expression.

$\frac{7}{10} + \frac{- 8 i}{5}$

Step 5.2

Move the negative in front of the fraction.

$\frac{7}{10} - \frac{8 i}{5}$