Basic Math Examples

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

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$27 + 3 \cdot 3^{2} (5 i) + 3 \cdot 3 {(5 i)}^{2} + {(5 i)}^{3}$

Step 2.1.2

Multiply $3$ by $3^{2}$ by adding the exponents.

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Step 2.1.2.1

Multiply $3$ by $3^{2}$ .

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Step 2.1.2.1.1

Raise $3$ to the power of $1$ .

$27 + 3^{1} \cdot 3^{2} (5 i) + 3 \cdot 3 {(5 i)}^{2} + {(5 i)}^{3}$

Step 2.1.2.1.2

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

$27 + 3^{1 + 2} (5 i) + 3 \cdot 3 {(5 i)}^{2} + {(5 i)}^{3}$

Step 2.1.2.2

Add $1$ and $2$ .

$27 + 3^{3} (5 i) + 3 \cdot 3 {(5 i)}^{2} + {(5 i)}^{3}$

Step 2.1.3

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

$27 + 27 (5 i) + 3 \cdot 3 {(5 i)}^{2} + {(5 i)}^{3}$

Step 2.1.4

Multiply $5$ by $27$ .

$27 + 135 i + 3 \cdot 3 {(5 i)}^{2} + {(5 i)}^{3}$

Step 2.1.5

Multiply $3$ by $3$ .

$27 + 135 i + 9 {(5 i)}^{2} + {(5 i)}^{3}$

Step 2.1.6

Apply the product rule to $5 i$ .

$27 + 135 i + 9 (5^{2} i^{2}) + {(5 i)}^{3}$

Step 2.1.7

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

$27 + 135 i + 9 (25 i^{2}) + {(5 i)}^{3}$

Step 2.1.8

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

$27 + 135 i + 9 (25 \cdot - 1) + {(5 i)}^{3}$

Step 2.1.9

Multiply $9 (25 \cdot - 1)$ .

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Step 2.1.9.1

Multiply $25$ by $- 1$ .

$27 + 135 i + 9 \cdot - 25 + {(5 i)}^{3}$

Step 2.1.9.2

Multiply $9$ by $- 25$ .

$27 + 135 i - 225 + {(5 i)}^{3}$

Step 2.1.10

Apply the product rule to $5 i$ .

$27 + 135 i - 225 + 5^{3} i^{3}$

Step 2.1.11

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

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

Step 2.1.12

Factor out $i^{2}$ .

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

Step 2.1.13

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

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

Step 2.1.14

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

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

Step 2.1.15

Multiply $- 1$ by $125$ .

$27 + 135 i - 225 - 125 i$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $225$ from $27$ .

$- 198 + 135 i - 125 i$

Step 2.2.2

Subtract $125 i$ from $135 i$ .

$- 198 + 10 i$