Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Rewrite $- 9$ as $- 1 (9)$ .

$- \frac{1}{2} \cdot (i^{3} (\sqrt{- 1 (9)} - 4)) - 3 i^{2}$

Step 1.2

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$- \frac{1}{2} \cdot (i^{3} (\sqrt{- 1} \cdot \sqrt{9} - 4)) - 3 i^{2}$

Step 1.3

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

$- \frac{1}{2} \cdot (i^{3} (i \cdot \sqrt{9} - 4)) - 3 i^{2}$

Step 1.4

Factor out $i^{2}$ .

$- \frac{1}{2} \cdot (i^{2} \cdot i (i \cdot \sqrt{9} - 4)) - 3 i^{2}$

Step 1.5

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

$- \frac{1}{2} \cdot (- 1 \cdot i (i \cdot \sqrt{9} - 4)) - 3 i^{2}$

Step 1.6

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

$- \frac{1}{2} \cdot (- i (i \cdot \sqrt{9} - 4)) - 3 i^{2}$

Step 1.7

Simplify each term.

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Step 1.7.1

Rewrite $9$ as $3^{2}$ .

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

Step 1.7.2

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

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

Step 1.7.3

Move $3$ to the left of $i$ .

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

Step 1.8

Apply the distributive property.

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

Step 1.9

Multiply $- i (3 i)$ .

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Step 1.9.1

Multiply $3$ by $- 1$ .

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

Step 1.9.2

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

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

Step 1.9.3

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

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

Step 1.9.4

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

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

Step 1.9.5

Add $1$ and $1$ .

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

Step 1.10

Multiply $- 4$ by $- 1$ .

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

Step 1.11

Simplify each term.

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Step 1.11.1

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

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

Step 1.11.2

Multiply $- 3$ by $- 1$ .

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

Step 1.12

Apply the distributive property.

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

Step 1.13

Multiply $- \frac{1}{2} \cdot 3$ .

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Step 1.13.1

Multiply $3$ by $- 1$ .

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

Step 1.13.2

Combine $- 3$ and $\frac{1}{2}$ .

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

Step 1.14

Cancel the common factor of $2$ .

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Step 1.14.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 1.14.2

Factor $2$ out of $4 i$ .

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

Step 1.14.3

Cancel the common factor.

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

Step 1.14.4

Rewrite the expression.

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

Step 1.15

Multiply $2$ by $- 1$ .

$\frac{- 3}{2} - 2 i - 3 i^{2}$

Step 1.16

Move the negative in front of the fraction.

$- \frac{3}{2} - 2 i - 3 i^{2}$

Step 1.17

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

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

Step 1.18

Multiply $- 3$ by $- 1$ .

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

Step 2

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3

Combine $3$ and $\frac{2}{2}$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Multiply $3$ by $2$ .

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

Step 5.2

Add $- 3$ and $6$ .

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