Precalculus Examples

$\frac{1}{3 - \sqrt{- 4}}$

Step 1

Pull out imaginary unit $i$ .

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Step 1.1

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{1}{3 - \sqrt{- 1 (4)}}$

Step 1.2

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$\frac{1}{3 - (\sqrt{- 1} \cdot \sqrt{4})}$

Step 1.3

Rewrite $\sqrt{- 1}$ as $i$ .

$\frac{1}{3 - (i \cdot \sqrt{4})}$

Step 2

Multiply the numerator and denominator of $\frac{1}{3 - 2 i}$ by the conjugate of $3 - 2 i$ to make the denominator real.

$\frac{1}{3 - 2 i} \cdot \frac{3 + 2 i}{3 + 2 i}$

Step 3

Multiply.

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Step 3.1

Combine.

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

Step 3.2

Multiply $3 + 2 i$ by $1$ .

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

Step 3.3

Simplify the denominator.

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Step 3.3.1

Expand $(3 - 2 i) (3 + 2 i)$ using the FOIL Method.

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Step 3.3.1.1

Apply the distributive property.

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

Step 3.3.1.2

Apply the distributive property.

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

Step 3.3.1.3

Apply the distributive property.

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

Step 3.3.2

Simplify.

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Step 3.3.2.1

Multiply $3$ by $3$ .

$\frac{3 + 2 i}{9 + 3 (2 i) - 2 i \cdot 3 - 2 i (2 i)}$

Step 3.3.2.2

Multiply $2$ by $3$ .

$\frac{3 + 2 i}{9 + 6 i - 2 i \cdot 3 - 2 i (2 i)}$

Step 3.3.2.3

Multiply $3$ by $- 2$ .

$\frac{3 + 2 i}{9 + 6 i - 6 i - 2 i (2 i)}$

Step 3.3.2.4

Multiply $2$ by $- 2$ .

$\frac{3 + 2 i}{9 + 6 i - 6 i - 4 i i}$

Step 3.3.2.5

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

$\frac{3 + 2 i}{9 + 6 i - 6 i - 4 (i^{1} i)}$

Step 3.3.2.6

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

$\frac{3 + 2 i}{9 + 6 i - 6 i - 4 (i^{1} i^{1})}$

Step 3.3.2.7

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

$\frac{3 + 2 i}{9 + 6 i - 6 i - 4 i^{1 + 1}}$

Step 3.3.2.8

Add $1$ and $1$ .

$\frac{3 + 2 i}{9 + 6 i - 6 i - 4 i^{2}}$

Step 3.3.2.9

Subtract $6 i$ from $6 i$ .

$\frac{3 + 2 i}{9 + 0 - 4 i^{2}}$

Step 3.3.2.10

Add $9$ and $0$ .

$\frac{3 + 2 i}{9 - 4 i^{2}}$

Step 3.3.3

Simplify each term.

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Step 3.3.3.1

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

$\frac{3 + 2 i}{9 - 4 \cdot - 1}$

Step 3.3.3.2

Multiply $- 4$ by $- 1$ .

$\frac{3 + 2 i}{9 + 4}$

Step 3.3.4

Add $9$ and $4$ .

$\frac{3 + 2 i}{13}$

Step 4

Split the fraction $\frac{3 + 2 i}{13}$ into two fractions.

$\frac{3}{13} + \frac{2 i}{13}$