Finite Math Examples

$\frac{0.5}{4 + 3 i}$

Step 1

Multiply the numerator and denominator of $\frac{0.5}{4 + 3 i}$ by the conjugate of $4 + 3 i$ to make the denominator real.

$\frac{0.5}{4 + 3 i} \cdot \frac{4 - 3 i}{4 - 3 i}$

Step 2

Multiply.

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Step 2.1

Combine.

$\frac{0.5 (4 - 3 i)}{(4 + 3 i) (4 - 3 i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the distributive property.

$\frac{0.5 \cdot 4 + 0.5 (- 3 i)}{(4 + 3 i) (4 - 3 i)}$

Step 2.2.2

Multiply $0.5$ by $4$ .

$\frac{2 + 0.5 (- 3 i)}{(4 + 3 i) (4 - 3 i)}$

Step 2.2.3

Multiply $- 3$ by $0.5$ .

$\frac{2 - 1.5 i}{(4 + 3 i) (4 - 3 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

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

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Step 2.3.1.1

Apply the distributive property.

$\frac{2 - 1.5 i}{4 (4 - 3 i) + 3 i (4 - 3 i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{2 - 1.5 i}{4 \cdot 4 + 4 (- 3 i) + 3 i (4 - 3 i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{2 - 1.5 i}{4 \cdot 4 + 4 (- 3 i) + 3 i \cdot 4 + 3 i (- 3 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $4$ by $4$ .

$\frac{2 - 1.5 i}{16 + 4 (- 3 i) + 3 i \cdot 4 + 3 i (- 3 i)}$

Step 2.3.2.2

Multiply $- 3$ by $4$ .

$\frac{2 - 1.5 i}{16 - 12 i + 3 i \cdot 4 + 3 i (- 3 i)}$

Step 2.3.2.3

Multiply $4$ by $3$ .

$\frac{2 - 1.5 i}{16 - 12 i + 12 i + 3 i (- 3 i)}$

Step 2.3.2.4

Multiply $- 3$ by $3$ .

$\frac{2 - 1.5 i}{16 - 12 i + 12 i - 9 i i}$

Step 2.3.2.5

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

$\frac{2 - 1.5 i}{16 - 12 i + 12 i - 9 (i^{1} i)}$

Step 2.3.2.6

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

$\frac{2 - 1.5 i}{16 - 12 i + 12 i - 9 (i^{1} i^{1})}$

Step 2.3.2.7

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

$\frac{2 - 1.5 i}{16 - 12 i + 12 i - 9 i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{2 - 1.5 i}{16 - 12 i + 12 i - 9 i^{2}}$

Step 2.3.2.9

Add $- 12 i$ and $12 i$ .

$\frac{2 - 1.5 i}{16 + 0 - 9 i^{2}}$

Step 2.3.2.10

Add $16$ and $0$ .

$\frac{2 - 1.5 i}{16 - 9 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{2 - 1.5 i}{16 - 9 \cdot - 1}$

Step 2.3.3.2

Multiply $- 9$ by $- 1$ .

$\frac{2 - 1.5 i}{16 + 9}$

Step 2.3.4

Add $16$ and $9$ .

$\frac{2 - 1.5 i}{25}$

Step 3

Split the fraction $\frac{2 - 1.5 i}{25}$ into two fractions.

$\frac{2}{25} + \frac{- 1.5 i}{25}$

Step 4

Cancel the common factor of $2$ and $25$ .

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Step 4.1

Rewrite $2$ as $1 (2)$ .

$\frac{1 (2)}{25} + \frac{- 1.5 i}{25}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Rewrite $25$ as $1 (25)$ .

$\frac{1 \cdot 2}{1 \cdot 25} + \frac{- 1.5 i}{25}$

Step 4.2.2

Cancel the common factor.

$\frac{1 \cdot 2}{1 \cdot 25} + \frac{- 1.5 i}{25}$

Step 4.2.3

Rewrite the expression.

$\frac{2}{25} + \frac{- 1.5 i}{25}$

Step 5

Move the negative in front of the fraction.

$\frac{2}{25} - \frac{1.5 i}{25}$

Step 6

Factor $1.5$ out of $1.5 i$ .

$\frac{2}{25} - \frac{1.5 (i)}{25}$

Step 7

Factor $25$ out of $25$ .

$\frac{2}{25} - \frac{1.5 (i)}{25 (1)}$

Step 8

Separate fractions.

$\frac{2}{25} - (\frac{1.5}{25} \cdot \frac{i}{1})$

Step 9

Divide $1.5$ by $25$ .

$\frac{2}{25} - (0.06 \frac{i}{1})$

Step 10

Divide $i$ by $1$ .

$\frac{2}{25} - (0.06 i)$

Step 11

Multiply $0.06$ by $- 1$ .

$\frac{2}{25} - 0.06 i$