Precalculus Examples

$(9 + 5 i) (9 + 8 i)$

Step 1

Expand $(9 + 5 i) (9 + 8 i)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$9 (9 + 8 i) + 5 i (9 + 8 i)$

Step 1.2

Apply the distributive property.

$9 \cdot 9 + 9 (8 i) + 5 i (9 + 8 i)$

Step 1.3

Apply the distributive property.

$9 \cdot 9 + 9 (8 i) + 5 i \cdot 9 + 5 i (8 i)$

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 $9$ by $9$ .

$81 + 9 (8 i) + 5 i \cdot 9 + 5 i (8 i)$

Step 2.1.2

Multiply $8$ by $9$ .

$81 + 72 i + 5 i \cdot 9 + 5 i (8 i)$

Step 2.1.3

Multiply $9$ by $5$ .

$81 + 72 i + 45 i + 5 i (8 i)$

Step 2.1.4

Multiply $5 i (8 i)$ .

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Step 2.1.4.1

Multiply $8$ by $5$ .

$81 + 72 i + 45 i + 40 i i$

Step 2.1.4.2

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

$81 + 72 i + 45 i + 40 (i^{1} i)$

Step 2.1.4.3

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

$81 + 72 i + 45 i + 40 (i^{1} i^{1})$

Step 2.1.4.4

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

$81 + 72 i + 45 i + 40 i^{1 + 1}$

Step 2.1.4.5

Add $1$ and $1$ .

$81 + 72 i + 45 i + 40 i^{2}$

Step 2.1.5

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

$81 + 72 i + 45 i + 40 \cdot - 1$

Step 2.1.6

Multiply $40$ by $- 1$ .

$81 + 72 i + 45 i - 40$

Step 2.2

Subtract $40$ from $81$ .

$41 + 72 i + 45 i$

Step 2.3

Add $72 i$ and $45 i$ .

$41 + 117 i$