Finite Math Examples

$\frac{60 i}{3 + i}$

Step 1

Multiply the numerator and denominator of $\frac{60 i}{3 + 1 i}$ by the conjugate of $3 + 1 i$ to make the denominator real.

$\frac{60 i}{3 + 1 i} \cdot \frac{3 - i}{3 - i}$

Step 2

Multiply.

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Step 2.1

Combine.

$\frac{60 i (3 - i)}{(3 + 1 i) (3 - i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the distributive property.

$\frac{60 i \cdot 3 + 60 i (- i)}{(3 + 1 i) (3 - i)}$

Step 2.2.2

Multiply $3$ by $60$ .

$\frac{180 i + 60 i (- i)}{(3 + 1 i) (3 - i)}$

Step 2.2.3

Multiply $60 i (- i)$ .

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Step 2.2.3.1

Multiply $- 1$ by $60$ .

$\frac{180 i - 60 i i}{(3 + 1 i) (3 - i)}$

Step 2.2.3.2

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

$\frac{180 i - 60 (i^{1} i)}{(3 + 1 i) (3 - i)}$

Step 2.2.3.3

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

$\frac{180 i - 60 (i^{1} i^{1})}{(3 + 1 i) (3 - i)}$

Step 2.2.3.4

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

$\frac{180 i - 60 i^{1 + 1}}{(3 + 1 i) (3 - i)}$

Step 2.2.3.5

Add $1$ and $1$ .

$\frac{180 i - 60 i^{2}}{(3 + 1 i) (3 - i)}$

Step 2.2.4

Simplify each term.

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Step 2.2.4.1

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

$\frac{180 i - 60 \cdot - 1}{(3 + 1 i) (3 - i)}$

Step 2.2.4.2

Multiply $- 60$ by $- 1$ .

$\frac{180 i + 60}{(3 + 1 i) (3 - i)}$

Step 2.2.5

Reorder $180 i$ and $60$ .

$\frac{60 + 180 i}{(3 + 1 i) (3 - i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand $(3 + 1 i) (3 - i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

$\frac{60 + 180 i}{3 (3 - i) + 1 i (3 - i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{60 + 180 i}{3 \cdot 3 + 3 (- i) + 1 i (3 - i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{60 + 180 i}{3 \cdot 3 + 3 (- i) + 1 i \cdot 3 + 1 i (- i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $3$ by $3$ .

$\frac{60 + 180 i}{9 + 3 (- i) + 1 i \cdot 3 + 1 i (- i)}$

Step 2.3.2.2

Multiply $- 1$ by $3$ .

$\frac{60 + 180 i}{9 - 3 i + 1 i \cdot 3 + 1 i (- i)}$

Step 2.3.2.3

Multiply $3$ by $1$ .

$\frac{60 + 180 i}{9 - 3 i + 3 i + 1 i (- i)}$

Step 2.3.2.4

Multiply $- 1$ by $1$ .

$\frac{60 + 180 i}{9 - 3 i + 3 i - i i}$

Step 2.3.2.5

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

$\frac{60 + 180 i}{9 - 3 i + 3 i - (i^{1} i)}$

Step 2.3.2.6

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

$\frac{60 + 180 i}{9 - 3 i + 3 i - (i^{1} i^{1})}$

Step 2.3.2.7

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

$\frac{60 + 180 i}{9 - 3 i + 3 i - i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{60 + 180 i}{9 - 3 i + 3 i - i^{2}}$

Step 2.3.2.9

Add $- 3 i$ and $3 i$ .

$\frac{60 + 180 i}{9 + 0 - i^{2}}$

Step 2.3.2.10

Add $9$ and $0$ .

$\frac{60 + 180 i}{9 - i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{60 + 180 i}{9 - - 1}$

Step 2.3.3.2

Multiply $- 1$ by $- 1$ .

$\frac{60 + 180 i}{9 + 1}$

Step 2.3.4

Add $9$ and $1$ .

$\frac{60 + 180 i}{10}$

Step 3

Cancel the common factor of $60 + 180 i$ and $10$ .

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Step 3.1

Factor $10$ out of $60$ .

$\frac{10 \cdot 6 + 180 i}{10}$

Step 3.2

Factor $10$ out of $180 i$ .

$\frac{10 \cdot 6 + 10 (18 i)}{10}$

Step 3.3

Factor $10$ out of $10 (6) + 10 (18 i)$ .

$\frac{10 (6 + 18 i)}{10}$

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor $10$ out of $10$ .

$\frac{10 (6 + 18 i)}{10 (1)}$

Step 3.4.2

Cancel the common factor.

$\frac{10 (6 + 18 i)}{10 \cdot 1}$

Step 3.4.3

Rewrite the expression.

$\frac{6 + 18 i}{1}$

Step 3.4.4

Divide $6 + 18 i$ by $1$ .

$6 + 18 i$