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

Add $0$ .

$| \frac{2 - 1.5 i}{25} + 0 |$

Step 4

Factor $i$ out of $\frac{2 - 1.5 i}{25} + 0$ .

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Step 4.1

Factor $i$ out of $\frac{2 - 1.5 i}{25}$ .

$| i \frac{- 2 i - 1.5}{25} + 0 |$

Step 4.2

Rewrite $0$ as $- 1 (0)$ .

$| i \frac{- 2 i - 1.5}{25} - 1 (0) |$

Step 4.3

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

$| i \frac{- 2 i - 1.5}{25} + i^{2} \cdot 0 |$

Step 4.4

Factor $i$ out of $i^{2} \cdot 0$ .

$| i \frac{- 2 i - 1.5}{25} + i (i \cdot 0) |$

Step 4.5

Reorder $i$ and $0$ .

$| i \frac{- 2 i - 1.5}{25} + i (0 i) |$

Step 4.6

Factor $i$ out of $i \frac{- 2 i - 1.5}{25} + i (0 i)$ .

$| i (\frac{- 2 i - 1.5}{25} + 0 i) |$

Step 4.7

Reorder $i$ and $\frac{- 2 i - 1.5}{25} + 0 i$ .

$| (\frac{- 2 i - 1.5}{25} + 0 i) i |$

Step 5

Use the formula $| a + b i | = \sqrt{a^{2} + b^{2}}$ to find the magnitude.

$\sqrt{0^{2} + {(\frac{- 2 i - 1.5}{25} + 0 i)}^{2}}$

Step 6

Raising $0$ to any positive power yields $0$ .

$\sqrt{0 + {(\frac{- 2 i - 1.5}{25} + 0 i)}^{2}}$

Step 7

Multiply $0$ by $i$ .

$\sqrt{0 + {(\frac{- 2 i - 1.5}{25} + 0)}^{2}}$

Step 8

Add $\frac{- 2 i - 1.5}{25}$ and $0$ .

$\sqrt{0 + {(\frac{- 2 i - 1.5}{25})}^{2}}$

Step 9

Apply the product rule to $\frac{- 2 i - 1.5}{25}$ .

$\sqrt{0 + \frac{{(- 2 i - 1.5)}^{2}}{25^{2}}}$

Step 10

Raise $25$ to the power of $2$ .

$\sqrt{0 + \frac{{(- 2 i - 1.5)}^{2}}{625}}$

Step 11

Add $0$ and $\frac{{(- 2 i - 1.5)}^{2}}{625}$ .

$\sqrt{\frac{{(- 2 i - 1.5)}^{2}}{625}}$

Step 12

Rewrite $\sqrt{\frac{{(- 2 i - 1.5)}^{2}}{625}}$ as $\frac{\sqrt{{(- 2 i - 1.5)}^{2}}}{\sqrt{625}}$ .

$\frac{\sqrt{{(- 2 i - 1.5)}^{2}}}{\sqrt{625}}$

Step 13

Simplify the numerator.

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Step 13.1

Pull terms out from under the radical, assuming positive real numbers.

$\frac{- (- 2 i - 1.5)}{\sqrt{625}}$

Step 13.2

Apply the distributive property.

$\frac{- (- 2 i) - - 1.5}{\sqrt{625}}$

Step 13.3

Multiply $- 2$ by $- 1$ .

$\frac{2 i - - 1.5}{\sqrt{625}}$

Step 13.4

Multiply $- 1$ by $- 1.5$ .

$\frac{2 i + 1.5}{\sqrt{625}}$

Step 14

Simplify the denominator.

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Step 14.1

Rewrite $625$ as $25^{2}$ .

$\frac{2 i + 1.5}{\sqrt{25^{2}}}$

Step 14.2

Pull terms out from under the radical, assuming positive real numbers.

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

Step 15

Reorder $2 i$ and $1.5$ .

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