Algebra Examples

${(x - 3 i)}^{3}$

Step 1

Use the Binomial Theorem.

$x^{3} + 3 x^{2} (- 3 i) + 3 x {(- 3 i)}^{2} + {(- 3 i)}^{3}$

Step 2

Simplify each term.

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Step 2.1

Multiply $- 3$ by $3$ .

$x^{3} - 9 x^{2} i + 3 x {(- 3 i)}^{2} + {(- 3 i)}^{3}$

Step 2.2

Apply the product rule to $- 3 i$ .

$x^{3} - 9 x^{2} i + 3 x ({(- 3)}^{2} i^{2}) + {(- 3 i)}^{3}$

Step 2.3

Raise $- 3$ to the power of $2$ .

$x^{3} - 9 x^{2} i + 3 x (9 i^{2}) + {(- 3 i)}^{3}$

Step 2.4

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

$x^{3} - 9 x^{2} i + 3 x (9 \cdot - 1) + {(- 3 i)}^{3}$

Step 2.5

Multiply $9$ by $- 1$ .

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

Step 2.6

Multiply $- 9$ by $3$ .

$x^{3} - 9 x^{2} i - 27 x + {(- 3 i)}^{3}$

Step 2.7

Apply the product rule to $- 3 i$ .

$x^{3} - 9 x^{2} i - 27 x + {(- 3)}^{3} i^{3}$

Step 2.8

Raise $- 3$ to the power of $3$ .

$x^{3} - 9 x^{2} i - 27 x - 27 i^{3}$

Step 2.9

Factor out $i^{2}$ .

$x^{3} - 9 x^{2} i - 27 x - 27 (i^{2} \cdot i)$

Step 2.10

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

$x^{3} - 9 x^{2} i - 27 x - 27 (- 1 \cdot i)$

Step 2.11

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

$x^{3} - 9 x^{2} i - 27 x - 27 (- i)$

Step 2.12

Multiply $- 1$ by $- 27$ .

$x^{3} - 9 x^{2} i - 27 x + 27 i$