Precalculus Examples

$(2 + 3 i) (3 - 2 i)$

Step 1

Expand $(2 + 3 i) (3 - 2 i)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$2 (3 - 2 i) + 3 i (3 - 2 i)$

Step 1.2

Apply the distributive property.

$2 \cdot 3 + 2 (- 2 i) + 3 i (3 - 2 i)$

Step 1.3

Apply the distributive property.

$2 \cdot 3 + 2 (- 2 i) + 3 i \cdot 3 + 3 i (- 2 i)$

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $2$ by $3$ .

$6 + 2 (- 2 i) + 3 i \cdot 3 + 3 i (- 2 i)$

Step 2.1.2

Multiply $- 2$ by $2$ .

$6 - 4 i + 3 i \cdot 3 + 3 i (- 2 i)$

Step 2.1.3

Multiply $3$ by $3$ .

$6 - 4 i + 9 i + 3 i (- 2 i)$

Step 2.1.4

Multiply $3 i (- 2 i)$ .

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Step 2.1.4.1

Multiply $- 2$ by $3$ .

$6 - 4 i + 9 i - 6 i i$

Step 2.1.4.2

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

$6 - 4 i + 9 i - 6 (i^{1} i)$

Step 2.1.4.3

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

$6 - 4 i + 9 i - 6 (i^{1} i^{1})$

Step 2.1.4.4

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

$6 - 4 i + 9 i - 6 i^{1 + 1}$

Step 2.1.4.5

Add $1$ and $1$ .

$6 - 4 i + 9 i - 6 i^{2}$

Step 2.1.5

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

$6 - 4 i + 9 i - 6 \cdot - 1$

Step 2.1.6

Multiply $- 6$ by $- 1$ .

$6 - 4 i + 9 i + 6$

Step 2.2

Add $6$ and $6$ .

$12 - 4 i + 9 i$

Step 2.3

Add $- 4 i$ and $9 i$ .

$12 + 5 i$