Precalculus Examples

$\frac{20 i}{1 - 3 i}$

Step 1

Multiply the numerator and denominator of $\frac{20 i}{1 - 3 i}$ by the conjugate of $1 - 3 i$ to make the denominator real.

$\frac{20 i}{1 - 3 i} \cdot \frac{1 + 3 i}{1 + 3 i}$

Step 2

Multiply.

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Step 2.1

Combine.

$\frac{20 i (1 + 3 i)}{(1 - 3 i) (1 + 3 i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the distributive property.

$\frac{20 i \cdot 1 + 20 i (3 i)}{(1 - 3 i) (1 + 3 i)}$

Step 2.2.2

Multiply $20$ by $1$ .

$\frac{20 i + 20 i (3 i)}{(1 - 3 i) (1 + 3 i)}$

Step 2.2.3

Multiply $20 i (3 i)$ .

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Step 2.2.3.1

Multiply $3$ by $20$ .

$\frac{20 i + 60 i i}{(1 - 3 i) (1 + 3 i)}$

Step 2.2.3.2

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

$\frac{20 i + 60 (i^{1} i)}{(1 - 3 i) (1 + 3 i)}$

Step 2.2.3.3

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

$\frac{20 i + 60 (i^{1} i^{1})}{(1 - 3 i) (1 + 3 i)}$

Step 2.2.3.4

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

$\frac{20 i + 60 i^{1 + 1}}{(1 - 3 i) (1 + 3 i)}$

Step 2.2.3.5

Add $1$ and $1$ .

$\frac{20 i + 60 i^{2}}{(1 - 3 i) (1 + 3 i)}$

Step 2.2.4

Simplify each term.

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Step 2.2.4.1

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

$\frac{20 i + 60 \cdot - 1}{(1 - 3 i) (1 + 3 i)}$

Step 2.2.4.2

Multiply $60$ by $- 1$ .

$\frac{20 i - 60}{(1 - 3 i) (1 + 3 i)}$

Step 2.2.5

Reorder $20 i$ and $- 60$ .

$\frac{- 60 + 20 i}{(1 - 3 i) (1 + 3 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

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

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Step 2.3.1.1

Apply the distributive property.

$\frac{- 60 + 20 i}{1 (1 + 3 i) - 3 i (1 + 3 i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{- 60 + 20 i}{1 \cdot 1 + 1 (3 i) - 3 i (1 + 3 i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{- 60 + 20 i}{1 \cdot 1 + 1 (3 i) - 3 i \cdot 1 - 3 i (3 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $1$ by $1$ .

$\frac{- 60 + 20 i}{1 + 1 (3 i) - 3 i \cdot 1 - 3 i (3 i)}$

Step 2.3.2.2

Multiply $3$ by $1$ .

$\frac{- 60 + 20 i}{1 + 3 i - 3 i \cdot 1 - 3 i (3 i)}$

Step 2.3.2.3

Multiply $- 3$ by $1$ .

$\frac{- 60 + 20 i}{1 + 3 i - 3 i - 3 i (3 i)}$

Step 2.3.2.4

Multiply $3$ by $- 3$ .

$\frac{- 60 + 20 i}{1 + 3 i - 3 i - 9 i i}$

Step 2.3.2.5

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

$\frac{- 60 + 20 i}{1 + 3 i - 3 i - 9 (i^{1} i)}$

Step 2.3.2.6

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

$\frac{- 60 + 20 i}{1 + 3 i - 3 i - 9 (i^{1} i^{1})}$

Step 2.3.2.7

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

$\frac{- 60 + 20 i}{1 + 3 i - 3 i - 9 i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{- 60 + 20 i}{1 + 3 i - 3 i - 9 i^{2}}$

Step 2.3.2.9

Subtract $3 i$ from $3 i$ .

$\frac{- 60 + 20 i}{1 + 0 - 9 i^{2}}$

Step 2.3.2.10

Add $1$ and $0$ .

$\frac{- 60 + 20 i}{1 - 9 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{- 60 + 20 i}{1 - 9 \cdot - 1}$

Step 2.3.3.2

Multiply $- 9$ by $- 1$ .

$\frac{- 60 + 20 i}{1 + 9}$

Step 2.3.4

Add $1$ and $9$ .

$\frac{- 60 + 20 i}{10}$

Step 3

Cancel the common factor of $- 60 + 20 i$ and $10$ .

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Step 3.1

Factor $10$ out of $- 60$ .

$\frac{10 \cdot - 6 + 20 i}{10}$

Step 3.2

Factor $10$ out of $20 i$ .

$\frac{10 \cdot - 6 + 10 (2 i)}{10}$

Step 3.3

Factor $10$ out of $10 (- 6) + 10 (2 i)$ .

$\frac{10 (- 6 + 2 i)}{10}$

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor $10$ out of $10$ .

$\frac{10 (- 6 + 2 i)}{10 (1)}$

Step 3.4.2

Cancel the common factor.

$\frac{10 (- 6 + 2 i)}{10 \cdot 1}$

Step 3.4.3

Rewrite the expression.

$\frac{- 6 + 2 i}{1}$

Step 3.4.4

Divide $- 6 + 2 i$ by $1$ .

$- 6 + 2 i$