Finite Math Examples

$(7 u^{2} + 8 u v - 6 v^{2}) (6 u^{2} + 4 u v + 3 v^{2})$

Step 1

Expand $(7 u^{2} + 8 u v - 6 v^{2}) (6 u^{2} + 4 u v + 3 v^{2})$ by multiplying each term in the first expression by each term in the second expression.

$7 u^{2} (6 u^{2}) + 7 u^{2} (4 u v) + 7 u^{2} (3 v^{2}) + 8 u v (6 u^{2}) + 8 u v (4 u v) + 8 u v (3 v^{2}) - 6 v^{2} (6 u^{2}) - 6 v^{2} (4 u v) - 6 v^{2} (3 v^{2})$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite using the commutative property of multiplication.

$7 \cdot 6 u^{2} u^{2} + 7 u^{2} (4 u v) + 7 u^{2} (3 v^{2}) + 8 u v (6 u^{2}) + 8 u v (4 u v) + 8 u v (3 v^{2}) - 6 v^{2} (6 u^{2}) - 6 v^{2} (4 u v) - 6 v^{2} (3 v^{2})$

Step 2.1.2

Multiply $u^{2}$ by $u^{2}$ by adding the exponents.

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Step 2.1.2.1

Move $u^{2}$ .

$7 \cdot 6 (u^{2} u^{2}) + 7 u^{2} (4 u v) + 7 u^{2} (3 v^{2}) + 8 u v (6 u^{2}) + 8 u v (4 u v) + 8 u v (3 v^{2}) - 6 v^{2} (6 u^{2}) - 6 v^{2} (4 u v) - 6 v^{2} (3 v^{2})$

Step 2.1.2.2

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

$7 \cdot 6 u^{2 + 2} + 7 u^{2} (4 u v) + 7 u^{2} (3 v^{2}) + 8 u v (6 u^{2}) + 8 u v (4 u v) + 8 u v (3 v^{2}) - 6 v^{2} (6 u^{2}) - 6 v^{2} (4 u v) - 6 v^{2} (3 v^{2})$

Step 2.1.2.3

Add $2$ and $2$ .

$7 \cdot 6 u^{4} + 7 u^{2} (4 u v) + 7 u^{2} (3 v^{2}) + 8 u v (6 u^{2}) + 8 u v (4 u v) + 8 u v (3 v^{2}) - 6 v^{2} (6 u^{2}) - 6 v^{2} (4 u v) - 6 v^{2} (3 v^{2})$

Step 2.1.3

Multiply $7$ by $6$ .

$42 u^{4} + 7 u^{2} (4 u v) + 7 u^{2} (3 v^{2}) + 8 u v (6 u^{2}) + 8 u v (4 u v) + 8 u v (3 v^{2}) - 6 v^{2} (6 u^{2}) - 6 v^{2} (4 u v) - 6 v^{2} (3 v^{2})$

Step 2.1.4

Multiply $u^{2}$ by $u$ by adding the exponents.

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Step 2.1.4.1

Move $u$ .

$42 u^{4} + 7 (u \cdot u^{2}) (4 v) + 7 u^{2} (3 v^{2}) + 8 u v (6 u^{2}) + 8 u v (4 u v) + 8 u v (3 v^{2}) - 6 v^{2} (6 u^{2}) - 6 v^{2} (4 u v) - 6 v^{2} (3 v^{2})$

Step 2.1.4.2

Multiply $u$ by $u^{2}$ .

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Step 2.1.4.2.1

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

$42 u^{4} + 7 (u^{1} u^{2}) (4 v) + 7 u^{2} (3 v^{2}) + 8 u v (6 u^{2}) + 8 u v (4 u v) + 8 u v (3 v^{2}) - 6 v^{2} (6 u^{2}) - 6 v^{2} (4 u v) - 6 v^{2} (3 v^{2})$

Step 2.1.4.2.2

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

$42 u^{4} + 7 u^{1 + 2} (4 v) + 7 u^{2} (3 v^{2}) + 8 u v (6 u^{2}) + 8 u v (4 u v) + 8 u v (3 v^{2}) - 6 v^{2} (6 u^{2}) - 6 v^{2} (4 u v) - 6 v^{2} (3 v^{2})$

Step 2.1.4.3

Add $1$ and $2$ .

$42 u^{4} + 7 u^{3} (4 v) + 7 u^{2} (3 v^{2}) + 8 u v (6 u^{2}) + 8 u v (4 u v) + 8 u v (3 v^{2}) - 6 v^{2} (6 u^{2}) - 6 v^{2} (4 u v) - 6 v^{2} (3 v^{2})$

Step 2.1.5

Rewrite using the commutative property of multiplication.

$42 u^{4} + 7 \cdot 4 u^{3} v + 7 u^{2} (3 v^{2}) + 8 u v (6 u^{2}) + 8 u v (4 u v) + 8 u v (3 v^{2}) - 6 v^{2} (6 u^{2}) - 6 v^{2} (4 u v) - 6 v^{2} (3 v^{2})$

Step 2.1.6

Multiply $7$ by $4$ .

$42 u^{4} + 28 u^{3} v + 7 u^{2} (3 v^{2}) + 8 u v (6 u^{2}) + 8 u v (4 u v) + 8 u v (3 v^{2}) - 6 v^{2} (6 u^{2}) - 6 v^{2} (4 u v) - 6 v^{2} (3 v^{2})$

Step 2.1.7

Rewrite using the commutative property of multiplication.

$42 u^{4} + 28 u^{3} v + 7 \cdot 3 u^{2} v^{2} + 8 u v (6 u^{2}) + 8 u v (4 u v) + 8 u v (3 v^{2}) - 6 v^{2} (6 u^{2}) - 6 v^{2} (4 u v) - 6 v^{2} (3 v^{2})$

Step 2.1.8

Multiply $7$ by $3$ .

$42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 8 u v (6 u^{2}) + 8 u v (4 u v) + 8 u v (3 v^{2}) - 6 v^{2} (6 u^{2}) - 6 v^{2} (4 u v) - 6 v^{2} (3 v^{2})$

Step 2.1.9

Multiply $u$ by $u^{2}$ by adding the exponents.

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Step 2.1.9.1

Move $u^{2}$ .

$42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 8 (u^{2} u) v \cdot 6 + 8 u v (4 u v) + 8 u v (3 v^{2}) - 6 v^{2} (6 u^{2}) - 6 v^{2} (4 u v) - 6 v^{2} (3 v^{2})$

Step 2.1.9.2

Multiply $u^{2}$ by $u$ .

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Step 2.1.9.2.1

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

$42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 8 (u^{2} u^{1}) v \cdot 6 + 8 u v (4 u v) + 8 u v (3 v^{2}) - 6 v^{2} (6 u^{2}) - 6 v^{2} (4 u v) - 6 v^{2} (3 v^{2})$

Step 2.1.9.2.2

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

$42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 8 u^{2 + 1} v \cdot 6 + 8 u v (4 u v) + 8 u v (3 v^{2}) - 6 v^{2} (6 u^{2}) - 6 v^{2} (4 u v) - 6 v^{2} (3 v^{2})$

Step 2.1.9.3

Add $2$ and $1$ .

$42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 8 u^{3} v \cdot 6 + 8 u v (4 u v) + 8 u v (3 v^{2}) - 6 v^{2} (6 u^{2}) - 6 v^{2} (4 u v) - 6 v^{2} (3 v^{2})$

Step 2.1.10

Multiply $6$ by $8$ .

$42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 48 u^{3} v + 8 u v (4 u v) + 8 u v (3 v^{2}) - 6 v^{2} (6 u^{2}) - 6 v^{2} (4 u v) - 6 v^{2} (3 v^{2})$

Step 2.1.11

Multiply $u$ by $u$ by adding the exponents.

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Step 2.1.11.1

Move $u$ .

$42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 48 u^{3} v + 8 (u \cdot u) v (4 v) + 8 u v (3 v^{2}) - 6 v^{2} (6 u^{2}) - 6 v^{2} (4 u v) - 6 v^{2} (3 v^{2})$

Step 2.1.11.2

Multiply $u$ by $u$ .

$42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 48 u^{3} v + 8 u^{2} v (4 v) + 8 u v (3 v^{2}) - 6 v^{2} (6 u^{2}) - 6 v^{2} (4 u v) - 6 v^{2} (3 v^{2})$

Step 2.1.12

Multiply $v$ by $v$ by adding the exponents.

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Step 2.1.12.1

Move $v$ .

$42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 48 u^{3} v + 8 u^{2} (v \cdot v) \cdot 4 + 8 u v (3 v^{2}) - 6 v^{2} (6 u^{2}) - 6 v^{2} (4 u v) - 6 v^{2} (3 v^{2})$

Step 2.1.12.2

Multiply $v$ by $v$ .

$42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 48 u^{3} v + 8 u^{2} v^{2} \cdot 4 + 8 u v (3 v^{2}) - 6 v^{2} (6 u^{2}) - 6 v^{2} (4 u v) - 6 v^{2} (3 v^{2})$

Step 2.1.13

Multiply $4$ by $8$ .

$42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 48 u^{3} v + 32 u^{2} v^{2} + 8 u v (3 v^{2}) - 6 v^{2} (6 u^{2}) - 6 v^{2} (4 u v) - 6 v^{2} (3 v^{2})$

Step 2.1.14

Multiply $v$ by $v^{2}$ by adding the exponents.

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Step 2.1.14.1

Move $v^{2}$ .

$42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 48 u^{3} v + 32 u^{2} v^{2} + 8 u (v^{2} v) \cdot 3 - 6 v^{2} (6 u^{2}) - 6 v^{2} (4 u v) - 6 v^{2} (3 v^{2})$

Step 2.1.14.2

Multiply $v^{2}$ by $v$ .

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Step 2.1.14.2.1

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

$42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 48 u^{3} v + 32 u^{2} v^{2} + 8 u (v^{2} v^{1}) \cdot 3 - 6 v^{2} (6 u^{2}) - 6 v^{2} (4 u v) - 6 v^{2} (3 v^{2})$

Step 2.1.14.2.2

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

$42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 48 u^{3} v + 32 u^{2} v^{2} + 8 u v^{2 + 1} \cdot 3 - 6 v^{2} (6 u^{2}) - 6 v^{2} (4 u v) - 6 v^{2} (3 v^{2})$

Step 2.1.14.3

Add $2$ and $1$ .

$42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 48 u^{3} v + 32 u^{2} v^{2} + 8 u v^{3} \cdot 3 - 6 v^{2} (6 u^{2}) - 6 v^{2} (4 u v) - 6 v^{2} (3 v^{2})$

Step 2.1.15

Multiply $3$ by $8$ .

$42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 48 u^{3} v + 32 u^{2} v^{2} + 24 u v^{3} - 6 v^{2} (6 u^{2}) - 6 v^{2} (4 u v) - 6 v^{2} (3 v^{2})$

Step 2.1.16

Rewrite using the commutative property of multiplication.

$42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 48 u^{3} v + 32 u^{2} v^{2} + 24 u v^{3} - 6 \cdot 6 v^{2} u^{2} - 6 v^{2} (4 u v) - 6 v^{2} (3 v^{2})$

Step 2.1.17

Multiply $- 6$ by $6$ .

$42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 48 u^{3} v + 32 u^{2} v^{2} + 24 u v^{3} - 36 v^{2} u^{2} - 6 v^{2} (4 u v) - 6 v^{2} (3 v^{2})$

Step 2.1.18

Multiply $v^{2}$ by $v$ by adding the exponents.

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Step 2.1.18.1

Move $v$ .

$42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 48 u^{3} v + 32 u^{2} v^{2} + 24 u v^{3} - 36 v^{2} u^{2} - 6 (v \cdot v^{2}) (4 u) - 6 v^{2} (3 v^{2})$

Step 2.1.18.2

Multiply $v$ by $v^{2}$ .

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Step 2.1.18.2.1

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

$42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 48 u^{3} v + 32 u^{2} v^{2} + 24 u v^{3} - 36 v^{2} u^{2} - 6 (v^{1} v^{2}) (4 u) - 6 v^{2} (3 v^{2})$

Step 2.1.18.2.2

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

$42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 48 u^{3} v + 32 u^{2} v^{2} + 24 u v^{3} - 36 v^{2} u^{2} - 6 v^{1 + 2} (4 u) - 6 v^{2} (3 v^{2})$

Step 2.1.18.3

Add $1$ and $2$ .

$42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 48 u^{3} v + 32 u^{2} v^{2} + 24 u v^{3} - 36 v^{2} u^{2} - 6 v^{3} (4 u) - 6 v^{2} (3 v^{2})$

Step 2.1.19

Rewrite using the commutative property of multiplication.

$42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 48 u^{3} v + 32 u^{2} v^{2} + 24 u v^{3} - 36 v^{2} u^{2} - 6 \cdot 4 v^{3} u - 6 v^{2} (3 v^{2})$

Step 2.1.20

Multiply $- 6$ by $4$ .

$42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 48 u^{3} v + 32 u^{2} v^{2} + 24 u v^{3} - 36 v^{2} u^{2} - 24 v^{3} u - 6 v^{2} (3 v^{2})$

Step 2.1.21

Rewrite using the commutative property of multiplication.

$42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 48 u^{3} v + 32 u^{2} v^{2} + 24 u v^{3} - 36 v^{2} u^{2} - 24 v^{3} u - 6 \cdot 3 v^{2} v^{2}$

Step 2.1.22

Multiply $v^{2}$ by $v^{2}$ by adding the exponents.

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Step 2.1.22.1

Move $v^{2}$ .

$42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 48 u^{3} v + 32 u^{2} v^{2} + 24 u v^{3} - 36 v^{2} u^{2} - 24 v^{3} u - 6 \cdot 3 (v^{2} v^{2})$

Step 2.1.22.2

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

$42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 48 u^{3} v + 32 u^{2} v^{2} + 24 u v^{3} - 36 v^{2} u^{2} - 24 v^{3} u - 6 \cdot 3 v^{2 + 2}$

Step 2.1.22.3

Add $2$ and $2$ .

$42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 48 u^{3} v + 32 u^{2} v^{2} + 24 u v^{3} - 36 v^{2} u^{2} - 24 v^{3} u - 6 \cdot 3 v^{4}$

Step 2.1.23

Multiply $- 6$ by $3$ .

$42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 48 u^{3} v + 32 u^{2} v^{2} + 24 u v^{3} - 36 v^{2} u^{2} - 24 v^{3} u - 18 v^{4}$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Combine the opposite terms in $42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 48 u^{3} v + 32 u^{2} v^{2} + 24 u v^{3} - 36 v^{2} u^{2} - 24 v^{3} u - 18 v^{4}$ .

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Step 2.2.1.1

Reorder the factors in the terms $24 u v^{3}$ and $- 24 v^{3} u$ .

$42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 48 u^{3} v + 32 u^{2} v^{2} + 24 v^{3} u - 36 v^{2} u^{2} - 24 v^{3} u - 18 v^{4}$

Step 2.2.1.2

Subtract $24 v^{3} u$ from $24 v^{3} u$ .

$42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 48 u^{3} v + 32 u^{2} v^{2} - 36 v^{2} u^{2} + 0 - 18 v^{4}$

Step 2.2.1.3

Add $42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 48 u^{3} v + 32 u^{2} v^{2} - 36 v^{2} u^{2}$ and $0$ .

$42 u^{4} + 28 u^{3} v + 21 u^{2} v^{2} + 48 u^{3} v + 32 u^{2} v^{2} - 36 v^{2} u^{2} - 18 v^{4}$

Step 2.2.2

Add $28 u^{3} v$ and $48 u^{3} v$ .

$42 u^{4} + 76 u^{3} v + 21 u^{2} v^{2} + 32 u^{2} v^{2} - 36 v^{2} u^{2} - 18 v^{4}$

Step 2.2.3

Add $21 u^{2} v^{2}$ and $32 u^{2} v^{2}$ .

$42 u^{4} + 76 u^{3} v + 53 u^{2} v^{2} - 36 v^{2} u^{2} - 18 v^{4}$

Step 3

Subtract $36 v^{2} u^{2}$ from $53 u^{2} v^{2}$ .

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Step 3.1

Move $v^{2}$ .

$42 u^{4} + 76 u^{3} v + 53 u^{2} v^{2} - 36 u^{2} v^{2} - 18 v^{4}$

Step 3.2

Subtract $36 u^{2} v^{2}$ from $53 u^{2} v^{2}$ .

$42 u^{4} + 76 u^{3} v + 17 u^{2} v^{2} - 18 v^{4}$