Basic Math Examples

$(3 - 5 i) (7 - i) - (9 - i) (9 + i)$

Step 1

Simplify each term.

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Step 1.1

Expand $(3 - 5 i) (7 - i)$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

$3 (7 - i) - 5 i (7 - i) - (9 - i) (9 + i)$

Step 1.1.2

Apply the distributive property.

$3 \cdot 7 + 3 (- i) - 5 i (7 - i) - (9 - i) (9 + i)$

Step 1.1.3

Apply the distributive property.

$3 \cdot 7 + 3 (- i) - 5 i \cdot 7 - 5 i (- i) - (9 - i) (9 + i)$

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 $3$ by $7$ .

$21 + 3 (- i) - 5 i \cdot 7 - 5 i (- i) - (9 - i) (9 + i)$

Step 1.2.1.2

Multiply $- 1$ by $3$ .

$21 - 3 i - 5 i \cdot 7 - 5 i (- i) - (9 - i) (9 + i)$

Step 1.2.1.3

Multiply $7$ by $- 5$ .

$21 - 3 i - 35 i - 5 i (- i) - (9 - i) (9 + i)$

Step 1.2.1.4

Multiply $- 5 i (- i)$ .

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Step 1.2.1.4.1

Multiply $- 1$ by $- 5$ .

$21 - 3 i - 35 i + 5 i i - (9 - i) (9 + i)$

Step 1.2.1.4.2

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

$21 - 3 i - 35 i + 5 (i^{1} i) - (9 - i) (9 + i)$

Step 1.2.1.4.3

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

$21 - 3 i - 35 i + 5 (i^{1} i^{1}) - (9 - i) (9 + i)$

Step 1.2.1.4.4

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

$21 - 3 i - 35 i + 5 i^{1 + 1} - (9 - i) (9 + i)$

Step 1.2.1.4.5

Add $1$ and $1$ .

$21 - 3 i - 35 i + 5 i^{2} - (9 - i) (9 + i)$

Step 1.2.1.5

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

$21 - 3 i - 35 i + 5 \cdot - 1 - (9 - i) (9 + i)$

Step 1.2.1.6

Multiply $5$ by $- 1$ .

$21 - 3 i - 35 i - 5 - (9 - i) (9 + i)$

Step 1.2.2

Subtract $5$ from $21$ .

$16 - 3 i - 35 i - (9 - i) (9 + i)$

Step 1.2.3

Subtract $35 i$ from $- 3 i$ .

$16 - 38 i - (9 - i) (9 + i)$

Step 1.3

Apply the distributive property.

$16 - 38 i + (- 1 \cdot 9 - - i) (9 + i)$

Step 1.4

Multiply $- 1$ by $9$ .

$16 - 38 i + (- 9 - - i) (9 + i)$

Step 1.5

Multiply $- 1$ by $- 1$ .

$16 - 38 i + (- 9 + 1 i) (9 + i)$

Step 1.6

Multiply $i$ by $1$ .

$16 - 38 i + (- 9 + i) (9 + i)$

Step 1.7

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

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Step 1.7.1

Apply the distributive property.

$16 - 38 i - 9 (9 + i) + i (9 + i)$

Step 1.7.2

Apply the distributive property.

$16 - 38 i - 9 \cdot 9 - 9 i + i (9 + i)$

Step 1.7.3

Apply the distributive property.

$16 - 38 i - 9 \cdot 9 - 9 i + i \cdot 9 + i i$

Step 1.8

Simplify and combine like terms.

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

Multiply $- 9$ by $9$ .

$16 - 38 i - 81 - 9 i + i \cdot 9 + i i$

Step 1.8.1.2

Move $9$ to the left of $i$ .

$16 - 38 i - 81 - 9 i + 9 \cdot i + i i$

Step 1.8.1.3

Multiply $i i$ .

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Step 1.8.1.3.1

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

$16 - 38 i - 81 - 9 i + 9 i + i^{1} i$

Step 1.8.1.3.2

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

$16 - 38 i - 81 - 9 i + 9 i + i^{1} i^{1}$

Step 1.8.1.3.3

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

$16 - 38 i - 81 - 9 i + 9 i + i^{1 + 1}$

Step 1.8.1.3.4

Add $1$ and $1$ .

$16 - 38 i - 81 - 9 i + 9 i + i^{2}$

Step 1.8.1.4

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

$16 - 38 i - 81 - 9 i + 9 i - 1$

Step 1.8.2

Subtract $1$ from $- 81$ .

$16 - 38 i - 82 - 9 i + 9 i$

Step 1.8.3

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

$16 - 38 i - 82 + 0$

Step 1.8.4

Add $- 82$ and $0$ .

$16 - 38 i - 82$

Step 2

Subtract $82$ from $16$ .

$- 66 - 38 i$