Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Expand $(4 - i) (- 3 + 7 i)$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

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

Step 1.1.2

Apply the distributive property.

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

Step 1.1.3

Apply the distributive property.

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

Step 1.2

Simplify and combine like terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Multiply $4$ by $- 3$ .

$- 12 + 4 (7 i) - i \cdot - 3 - i (7 i) - 7 i (8 + 2 i)$

Step 1.2.1.2

Multiply $7$ by $4$ .

$- 12 + 28 i - i \cdot - 3 - i (7 i) - 7 i (8 + 2 i)$

Step 1.2.1.3

Multiply $- 3$ by $- 1$ .

$- 12 + 28 i + 3 i - i (7 i) - 7 i (8 + 2 i)$

Step 1.2.1.4

Multiply $- i (7 i)$ .

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Step 1.2.1.4.1

Multiply $7$ by $- 1$ .

$- 12 + 28 i + 3 i - 7 i i - 7 i (8 + 2 i)$

Step 1.2.1.4.2

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

$- 12 + 28 i + 3 i - 7 (i^{1} i) - 7 i (8 + 2 i)$

Step 1.2.1.4.3

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

$- 12 + 28 i + 3 i - 7 (i^{1} i^{1}) - 7 i (8 + 2 i)$

Step 1.2.1.4.4

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

$- 12 + 28 i + 3 i - 7 i^{1 + 1} - 7 i (8 + 2 i)$

Step 1.2.1.4.5

Add $1$ and $1$ .

$- 12 + 28 i + 3 i - 7 i^{2} - 7 i (8 + 2 i)$

Step 1.2.1.5

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

$- 12 + 28 i + 3 i - 7 \cdot - 1 - 7 i (8 + 2 i)$

Step 1.2.1.6

Multiply $- 7$ by $- 1$ .

$- 12 + 28 i + 3 i + 7 - 7 i (8 + 2 i)$

Step 1.2.2

Add $- 12$ and $7$ .

$- 5 + 28 i + 3 i - 7 i (8 + 2 i)$

Step 1.2.3

Add $28 i$ and $3 i$ .

$- 5 + 31 i - 7 i (8 + 2 i)$

Step 1.3

Apply the distributive property.

$- 5 + 31 i - 7 i \cdot 8 - 7 i (2 i)$

Step 1.4

Multiply $8$ by $- 7$ .

$- 5 + 31 i - 56 i - 7 i (2 i)$

Step 1.5

Multiply $- 7 i (2 i)$ .

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Step 1.5.1

Multiply $2$ by $- 7$ .

$- 5 + 31 i - 56 i - 14 i i$

Step 1.5.2

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

$- 5 + 31 i - 56 i - 14 (i^{1} i)$

Step 1.5.3

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

$- 5 + 31 i - 56 i - 14 (i^{1} i^{1})$

Step 1.5.4

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

$- 5 + 31 i - 56 i - 14 i^{1 + 1}$

Step 1.5.5

Add $1$ and $1$ .

$- 5 + 31 i - 56 i - 14 i^{2}$

Step 1.6

Simplify each term.

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Step 1.6.1

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

$- 5 + 31 i - 56 i - 14 \cdot - 1$

Step 1.6.2

Multiply $- 14$ by $- 1$ .

$- 5 + 31 i - 56 i + 14$

Step 2

Simplify by adding terms.

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Step 2.1

Add $- 5$ and $14$ .

$9 + 31 i - 56 i$

Step 2.2

Subtract $56 i$ from $31 i$ .

$9 - 25 i$