Algebra Examples

$(- 2 - 2 i) (- 4 - 3 i) (7 + 8 i)$

Step 1

Expand $(- 2 - 2 i) (- 4 - 3 i)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$(- 2 (- 4 - 3 i) - 2 i (- 4 - 3 i)) (7 + 8 i)$

Step 1.2

Apply the distributive property.

$(- 2 \cdot - 4 - 2 (- 3 i) - 2 i (- 4 - 3 i)) (7 + 8 i)$

Step 1.3

Apply the distributive property.

$(- 2 \cdot - 4 - 2 (- 3 i) - 2 i \cdot - 4 - 2 i (- 3 i)) (7 + 8 i)$

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 $- 2$ by $- 4$ .

$(8 - 2 (- 3 i) - 2 i \cdot - 4 - 2 i (- 3 i)) (7 + 8 i)$

Step 2.1.2

Multiply $- 3$ by $- 2$ .

$(8 + 6 i - 2 i \cdot - 4 - 2 i (- 3 i)) (7 + 8 i)$

Step 2.1.3

Multiply $- 4$ by $- 2$ .

$(8 + 6 i + 8 i - 2 i (- 3 i)) (7 + 8 i)$

Step 2.1.4

Multiply $- 2 i (- 3 i)$ .

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Step 2.1.4.1

Multiply $- 3$ by $- 2$ .

$(8 + 6 i + 8 i + 6 i i) (7 + 8 i)$

Step 2.1.4.2

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

$(8 + 6 i + 8 i + 6 (i^{1} i)) (7 + 8 i)$

Step 2.1.4.3

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

$(8 + 6 i + 8 i + 6 (i^{1} i^{1})) (7 + 8 i)$

Step 2.1.4.4

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

$(8 + 6 i + 8 i + 6 i^{1 + 1}) (7 + 8 i)$

Step 2.1.4.5

Add $1$ and $1$ .

$(8 + 6 i + 8 i + 6 i^{2}) (7 + 8 i)$

Step 2.1.5

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

$(8 + 6 i + 8 i + 6 \cdot - 1) (7 + 8 i)$

Step 2.1.6

Multiply $6$ by $- 1$ .

$(8 + 6 i + 8 i - 6) (7 + 8 i)$

Step 2.2

Subtract $6$ from $8$ .

$(2 + 6 i + 8 i) (7 + 8 i)$

Step 2.3

Add $6 i$ and $8 i$ .

$(2 + 14 i) (7 + 8 i)$

Step 3

Expand $(2 + 14 i) (7 + 8 i)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$2 (7 + 8 i) + 14 i (7 + 8 i)$

Step 3.2

Apply the distributive property.

$2 \cdot 7 + 2 (8 i) + 14 i (7 + 8 i)$

Step 3.3

Apply the distributive property.

$2 \cdot 7 + 2 (8 i) + 14 i \cdot 7 + 14 i (8 i)$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $2$ by $7$ .

$14 + 2 (8 i) + 14 i \cdot 7 + 14 i (8 i)$

Step 4.1.2

Multiply $8$ by $2$ .

$14 + 16 i + 14 i \cdot 7 + 14 i (8 i)$

Step 4.1.3

Multiply $7$ by $14$ .

$14 + 16 i + 98 i + 14 i (8 i)$

Step 4.1.4

Multiply $14 i (8 i)$ .

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Step 4.1.4.1

Multiply $8$ by $14$ .

$14 + 16 i + 98 i + 112 i i$

Step 4.1.4.2

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

$14 + 16 i + 98 i + 112 (i^{1} i)$

Step 4.1.4.3

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

$14 + 16 i + 98 i + 112 (i^{1} i^{1})$

Step 4.1.4.4

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

$14 + 16 i + 98 i + 112 i^{1 + 1}$

Step 4.1.4.5

Add $1$ and $1$ .

$14 + 16 i + 98 i + 112 i^{2}$

Step 4.1.5

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

$14 + 16 i + 98 i + 112 \cdot - 1$

Step 4.1.6

Multiply $112$ by $- 1$ .

$14 + 16 i + 98 i - 112$

Step 4.2

Subtract $112$ from $14$ .

$- 98 + 16 i + 98 i$

Step 4.3

Add $16 i$ and $98 i$ .

$- 98 + 114 i$