Algebra Examples

$(8 + 5 i) (3 + 2 i) - (4 + i) (4 - i)$

Step 1

Simplify each term.

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Step 1.1

Expand $(8 + 5 i) (3 + 2 i)$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

$8 (3 + 2 i) + 5 i (3 + 2 i) - (4 + i) (4 - i)$

Step 1.1.2

Apply the distributive property.

$8 \cdot 3 + 8 (2 i) + 5 i (3 + 2 i) - (4 + i) (4 - i)$

Step 1.1.3

Apply the distributive property.

$8 \cdot 3 + 8 (2 i) + 5 i \cdot 3 + 5 i (2 i) - (4 + i) (4 - i)$

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 $8$ by $3$ .

$24 + 8 (2 i) + 5 i \cdot 3 + 5 i (2 i) - (4 + i) (4 - i)$

Step 1.2.1.2

Multiply $2$ by $8$ .

$24 + 16 i + 5 i \cdot 3 + 5 i (2 i) - (4 + i) (4 - i)$

Step 1.2.1.3

Multiply $3$ by $5$ .

$24 + 16 i + 15 i + 5 i (2 i) - (4 + i) (4 - i)$

Step 1.2.1.4

Multiply $5 i (2 i)$ .

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Step 1.2.1.4.1

Multiply $2$ by $5$ .

$24 + 16 i + 15 i + 10 i i - (4 + i) (4 - i)$

Step 1.2.1.4.2

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

$24 + 16 i + 15 i + 10 (i^{1} i) - (4 + i) (4 - i)$

Step 1.2.1.4.3

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

$24 + 16 i + 15 i + 10 (i^{1} i^{1}) - (4 + i) (4 - i)$

Step 1.2.1.4.4

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

$24 + 16 i + 15 i + 10 i^{1 + 1} - (4 + i) (4 - i)$

Step 1.2.1.4.5

Add $1$ and $1$ .

$24 + 16 i + 15 i + 10 i^{2} - (4 + i) (4 - i)$

Step 1.2.1.5

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

$24 + 16 i + 15 i + 10 \cdot - 1 - (4 + i) (4 - i)$

Step 1.2.1.6

Multiply $10$ by $- 1$ .

$24 + 16 i + 15 i - 10 - (4 + i) (4 - i)$

Step 1.2.2

Subtract $10$ from $24$ .

$14 + 16 i + 15 i - (4 + i) (4 - i)$

Step 1.2.3

Add $16 i$ and $15 i$ .

$14 + 31 i - (4 + i) (4 - i)$

Step 1.3

Apply the distributive property.

$14 + 31 i + (- 1 \cdot 4 - i) (4 - i)$

Step 1.4

Multiply $- 1$ by $4$ .

$14 + 31 i + (- 4 - i) (4 - i)$

Step 1.5

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

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Step 1.5.1

Apply the distributive property.

$14 + 31 i - 4 (4 - i) - i (4 - i)$

Step 1.5.2

Apply the distributive property.

$14 + 31 i - 4 \cdot 4 - 4 (- i) - i (4 - i)$

Step 1.5.3

Apply the distributive property.

$14 + 31 i - 4 \cdot 4 - 4 (- i) - i \cdot 4 - i (- i)$

Step 1.6

Simplify and combine like terms.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

Multiply $- 4$ by $4$ .

$14 + 31 i - 16 - 4 (- i) - i \cdot 4 - i (- i)$

Step 1.6.1.2

Multiply $- 1$ by $- 4$ .

$14 + 31 i - 16 + 4 i - i \cdot 4 - i (- i)$

Step 1.6.1.3

Multiply $4$ by $- 1$ .

$14 + 31 i - 16 + 4 i - 4 i - i (- i)$

Step 1.6.1.4

Multiply $- i (- i)$ .

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Step 1.6.1.4.1

Multiply $- 1$ by $- 1$ .

$14 + 31 i - 16 + 4 i - 4 i + 1 i i$

Step 1.6.1.4.2

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

$14 + 31 i - 16 + 4 i - 4 i + 1 (i^{1} i)$

Step 1.6.1.4.3

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

$14 + 31 i - 16 + 4 i - 4 i + 1 (i^{1} i^{1})$

Step 1.6.1.4.4

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

$14 + 31 i - 16 + 4 i - 4 i + 1 i^{1 + 1}$

Step 1.6.1.4.5

Add $1$ and $1$ .

$14 + 31 i - 16 + 4 i - 4 i + 1 i^{2}$

Step 1.6.1.4.6

Multiply $i^{2}$ by $1$ .

$14 + 31 i - 16 + 4 i - 4 i + i^{2}$

Step 1.6.1.5

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

$14 + 31 i - 16 + 4 i - 4 i - 1$

Step 1.6.2

Subtract $1$ from $- 16$ .

$14 + 31 i - 17 + 4 i - 4 i$

Step 1.6.3

Subtract $4 i$ from $4 i$ .

$14 + 31 i - 17 + 0$

Step 1.6.4

Add $- 17$ and $0$ .

$14 + 31 i - 17$

Step 2

Subtract $17$ from $14$ .

$- 3 + 31 i$