Basic Math Examples

${(5 - 3 i)}^{3}$

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Raise $5$ to the power of $3$ .

$125 + 3 \cdot 5^{2} (- 3 i) + 3 \cdot 5 {(- 3 i)}^{2} + {(- 3 i)}^{3}$

Step 2.1.2

Raise $5$ to the power of $2$ .

$125 + 3 \cdot 25 (- 3 i) + 3 \cdot 5 {(- 3 i)}^{2} + {(- 3 i)}^{3}$

Step 2.1.3

Multiply $3$ by $25$ .

$125 + 75 (- 3 i) + 3 \cdot 5 {(- 3 i)}^{2} + {(- 3 i)}^{3}$

Step 2.1.4

Multiply $- 3$ by $75$ .

$125 - 225 i + 3 \cdot 5 {(- 3 i)}^{2} + {(- 3 i)}^{3}$

Step 2.1.5

Multiply $3$ by $5$ .

$125 - 225 i + 15 {(- 3 i)}^{2} + {(- 3 i)}^{3}$

Step 2.1.6

Apply the product rule to $- 3 i$ .

$125 - 225 i + 15 ({(- 3)}^{2} i^{2}) + {(- 3 i)}^{3}$

Step 2.1.7

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

$125 - 225 i + 15 (9 i^{2}) + {(- 3 i)}^{3}$

Step 2.1.8

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

$125 - 225 i + 15 (9 \cdot - 1) + {(- 3 i)}^{3}$

Step 2.1.9

Multiply $15 (9 \cdot - 1)$ .

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Step 2.1.9.1

Multiply $9$ by $- 1$ .

$125 - 225 i + 15 \cdot - 9 + {(- 3 i)}^{3}$

Step 2.1.9.2

Multiply $15$ by $- 9$ .

$125 - 225 i - 135 + {(- 3 i)}^{3}$

Step 2.1.10

Apply the product rule to $- 3 i$ .

$125 - 225 i - 135 + {(- 3)}^{3} i^{3}$

Step 2.1.11

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

$125 - 225 i - 135 - 27 i^{3}$

Step 2.1.12

Factor out $i^{2}$ .

$125 - 225 i - 135 - 27 (i^{2} \cdot i)$

Step 2.1.13

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

$125 - 225 i - 135 - 27 (- 1 \cdot i)$

Step 2.1.14

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

$125 - 225 i - 135 - 27 (- i)$

Step 2.1.15

Multiply $- 1$ by $- 27$ .

$125 - 225 i - 135 + 27 i$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $135$ from $125$ .

$- 10 - 225 i + 27 i$

Step 2.2.2

Add $- 225 i$ and $27 i$ .

$- 10 - 198 i$