Simplify (2+3i)^3

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

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify terms.

Simplify each term.

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

$8 + 3 \cdot 2^{2} (3 i) + 3 \cdot 2 {(3 i)}^{2} + {(3 i)}^{3}$

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

$8 + 3 \cdot 4 (3 i) + 3 \cdot 2 {(3 i)}^{2} + {(3 i)}^{3}$

Multiply $3$ by $4$ .

$8 + 12 (3 i) + 3 \cdot 2 {(3 i)}^{2} + {(3 i)}^{3}$

Multiply $3$ by $12$ .

$8 + 36 i + 3 \cdot 2 {(3 i)}^{2} + {(3 i)}^{3}$

Multiply $3$ by $2$ .

$8 + 36 i + 6 {(3 i)}^{2} + {(3 i)}^{3}$

Apply the product rule to $3 i$ .

$8 + 36 i + 6 (3^{2} i^{2}) + {(3 i)}^{3}$

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

$8 + 36 i + 6 (9 i^{2}) + {(3 i)}^{3}$

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

$8 + 36 i + 6 (9 \cdot - 1) + {(3 i)}^{3}$

Multiply $6 (9 \cdot - 1)$ .

Multiply $9$ by $- 1$ .

$8 + 36 i + 6 \cdot - 9 + {(3 i)}^{3}$

Multiply $6$ by $- 9$ .

$8 + 36 i - 54 + {(3 i)}^{3}$

Apply the product rule to $3 i$ .

$8 + 36 i - 54 + 3^{3} i^{3}$

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

$8 + 36 i - 54 + 27 i^{3}$

Factor out $i^{2}$ .

$8 + 36 i - 54 + 27 (i^{2} \cdot i)$

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

$8 + 36 i - 54 + 27 (- 1 \cdot i)$

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

$8 + 36 i - 54 + 27 (- i)$

Multiply $- 1$ by $27$ .

$8 + 36 i - 54 - 27 i$

Simplify by adding terms.

Subtract $54$ from $8$ .

$- 46 + 36 i - 27 i$

Subtract $27 i$ from $36 i$ .

$- 46 + 9 i$