Precalculus Examples

$\frac{3 i^{3} - 2 i^{2} - i^{5}}{2 - 4 i^{2} + \sqrt{- 4}}$

Step 1

Pull out imaginary unit $i$ .

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Step 1.1

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

$\frac{3 i^{3} - 2 i^{2} - i^{5}}{2 - 4 i^{2} + \sqrt{- 1 (4)}}$

Step 1.2

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

$\frac{3 i^{3} - 2 i^{2} - i^{5}}{2 - 4 i^{2} + \sqrt{- 1} \cdot \sqrt{4}}$

Step 1.3

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

$\frac{3 i^{3} - 2 i^{2} - i^{5}}{2 - 4 i^{2} + i \cdot \sqrt{4}}$

Step 2

Simplify the numerator.

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Step 2.1

Factor out $i^{2}$ .

$\frac{3 (i^{2} \cdot i) - 2 i^{2} - i^{5}}{2 - 4 i^{2} + i \cdot \sqrt{4}}$

Step 2.2

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

$\frac{3 (- 1 \cdot i) - 2 i^{2} - i^{5}}{2 - 4 i^{2} + i \cdot \sqrt{4}}$

Step 2.3

Rewrite $- 1 i$ as $- i$ .

$\frac{3 (- i) - 2 i^{2} - i^{5}}{2 - 4 i^{2} + i \cdot \sqrt{4}}$

Step 2.4

Multiply $- 1$ by $3$ .

$\frac{- 3 i - 2 i^{2} - i^{5}}{2 - 4 i^{2} + i \cdot \sqrt{4}}$

Step 2.5

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

$\frac{- 3 i - 2 \cdot - 1 - i^{5}}{2 - 4 i^{2} + i \cdot \sqrt{4}}$

Step 2.6

Multiply $- 2$ by $- 1$ .

$\frac{- 3 i + 2 - i^{5}}{2 - 4 i^{2} + i \cdot \sqrt{4}}$

Step 2.7

Factor out $i^{4}$ .

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

Step 2.8

Rewrite $i^{4}$ as $1$ .

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Step 2.8.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

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

Step 2.8.2

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

$\frac{- 3 i + 2 - ({(- 1)}^{2} i)}{2 - 4 i^{2} + i \cdot \sqrt{4}}$

Step 2.8.3

Raise $- 1$ to the power of $2$ .

$\frac{- 3 i + 2 - (1 i)}{2 - 4 i^{2} + i \cdot \sqrt{4}}$

Step 2.9

Multiply $i$ by $1$ .

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

Step 2.10

Subtract $i$ from $- 3 i$ .

$\frac{2 - 4 i}{2 - 4 i^{2} + i \cdot \sqrt{4}}$

Step 3

Simplify the denominator.

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Step 3.1

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

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

Step 3.2

Multiply $- 4$ by $- 1$ .

$\frac{2 - 4 i}{2 + 4 + i \cdot \sqrt{4}}$

Step 3.3

Rewrite $4$ as $2^{2}$ .

$\frac{2 - 4 i}{2 + 4 + i \cdot \sqrt{2^{2}}}$

Step 3.4

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

$\frac{2 - 4 i}{2 + 4 + i \cdot 2}$

Step 3.5

Move $2$ to the left of $i$ .

$\frac{2 - 4 i}{2 + 4 + 2 \cdot i}$

Step 3.6

Add $2$ and $4$ .

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

Step 4

Cancel the common factor of $2 - 4 i$ and $6 + 2 i$ .

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Step 4.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 1 - 4 i}{6 + 2 i}$

Step 4.2

Factor $2$ out of $- 4 i$ .

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

Step 4.3

Factor $2$ out of $2 (1) + 2 (- 2 i)$ .

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

Step 4.4

Cancel the common factors.

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Step 4.4.1

Factor $2$ out of $6$ .

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

Step 4.4.2

Factor $2$ out of $2 i$ .

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

Step 4.4.3

Factor $2$ out of $2 (3) + 2 (i)$ .

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

Step 4.4.4

Cancel the common factor.

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

Step 4.4.5

Rewrite the expression.

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