Basic Math Examples

$(2 u + 6 v + 5) (5 u - 7 v - 7)$

Step 1

Expand $(2 u + 6 v + 5) (5 u - 7 v - 7)$ by multiplying each term in the first expression by each term in the second expression.

$2 u (5 u) + 2 u (- 7 v) + 2 u \cdot - 7 + 6 v (5 u) + 6 v (- 7 v) + 6 v \cdot - 7 + 5 (5 u) + 5 (- 7 v) + 5 \cdot - 7$

Step 2

Simplify each term.

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Step 2.1

Rewrite using the commutative property of multiplication.

$2 \cdot 5 u \cdot u + 2 u (- 7 v) + 2 u \cdot - 7 + 6 v (5 u) + 6 v (- 7 v) + 6 v \cdot - 7 + 5 (5 u) + 5 (- 7 v) + 5 \cdot - 7$

Step 2.2

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

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Step 2.2.1

Move $u$ .

$2 \cdot 5 (u \cdot u) + 2 u (- 7 v) + 2 u \cdot - 7 + 6 v (5 u) + 6 v (- 7 v) + 6 v \cdot - 7 + 5 (5 u) + 5 (- 7 v) + 5 \cdot - 7$

Step 2.2.2

Multiply $u$ by $u$ .

$2 \cdot 5 u^{2} + 2 u (- 7 v) + 2 u \cdot - 7 + 6 v (5 u) + 6 v (- 7 v) + 6 v \cdot - 7 + 5 (5 u) + 5 (- 7 v) + 5 \cdot - 7$

Step 2.3

Multiply $2$ by $5$ .

$10 u^{2} + 2 u (- 7 v) + 2 u \cdot - 7 + 6 v (5 u) + 6 v (- 7 v) + 6 v \cdot - 7 + 5 (5 u) + 5 (- 7 v) + 5 \cdot - 7$

Step 2.4

Rewrite using the commutative property of multiplication.

$10 u^{2} + 2 \cdot - 7 u v + 2 u \cdot - 7 + 6 v (5 u) + 6 v (- 7 v) + 6 v \cdot - 7 + 5 (5 u) + 5 (- 7 v) + 5 \cdot - 7$

Step 2.5

Multiply $2$ by $- 7$ .

$10 u^{2} - 14 u v + 2 u \cdot - 7 + 6 v (5 u) + 6 v (- 7 v) + 6 v \cdot - 7 + 5 (5 u) + 5 (- 7 v) + 5 \cdot - 7$

Step 2.6

Multiply $- 7$ by $2$ .

$10 u^{2} - 14 u v - 14 u + 6 v (5 u) + 6 v (- 7 v) + 6 v \cdot - 7 + 5 (5 u) + 5 (- 7 v) + 5 \cdot - 7$

Step 2.7

Rewrite using the commutative property of multiplication.

$10 u^{2} - 14 u v - 14 u + 6 \cdot 5 v u + 6 v (- 7 v) + 6 v \cdot - 7 + 5 (5 u) + 5 (- 7 v) + 5 \cdot - 7$

Step 2.8

Multiply $6$ by $5$ .

$10 u^{2} - 14 u v - 14 u + 30 v u + 6 v (- 7 v) + 6 v \cdot - 7 + 5 (5 u) + 5 (- 7 v) + 5 \cdot - 7$

Step 2.9

Rewrite using the commutative property of multiplication.

$10 u^{2} - 14 u v - 14 u + 30 v u + 6 \cdot - 7 v \cdot v + 6 v \cdot - 7 + 5 (5 u) + 5 (- 7 v) + 5 \cdot - 7$

Step 2.10

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

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Step 2.10.1

Move $v$ .

$10 u^{2} - 14 u v - 14 u + 30 v u + 6 \cdot - 7 (v \cdot v) + 6 v \cdot - 7 + 5 (5 u) + 5 (- 7 v) + 5 \cdot - 7$

Step 2.10.2

Multiply $v$ by $v$ .

$10 u^{2} - 14 u v - 14 u + 30 v u + 6 \cdot - 7 v^{2} + 6 v \cdot - 7 + 5 (5 u) + 5 (- 7 v) + 5 \cdot - 7$

Step 2.11

Multiply $6$ by $- 7$ .

$10 u^{2} - 14 u v - 14 u + 30 v u - 42 v^{2} + 6 v \cdot - 7 + 5 (5 u) + 5 (- 7 v) + 5 \cdot - 7$

Step 2.12

Multiply $- 7$ by $6$ .

$10 u^{2} - 14 u v - 14 u + 30 v u - 42 v^{2} - 42 v + 5 (5 u) + 5 (- 7 v) + 5 \cdot - 7$

Step 2.13

Multiply $5$ by $5$ .

$10 u^{2} - 14 u v - 14 u + 30 v u - 42 v^{2} - 42 v + 25 u + 5 (- 7 v) + 5 \cdot - 7$

Step 2.14

Multiply $- 7$ by $5$ .

$10 u^{2} - 14 u v - 14 u + 30 v u - 42 v^{2} - 42 v + 25 u - 35 v + 5 \cdot - 7$

Step 2.15

Multiply $5$ by $- 7$ .

$10 u^{2} - 14 u v - 14 u + 30 v u - 42 v^{2} - 42 v + 25 u - 35 v - 35$

Step 3

Add $- 14 u v$ and $30 v u$ .

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Step 3.1

Move $v$ .

$10 u^{2} - 14 u v + 30 u v - 14 u - 42 v^{2} - 42 v + 25 u - 35 v - 35$

Step 3.2

Add $- 14 u v$ and $30 u v$ .

$10 u^{2} + 16 u v - 14 u - 42 v^{2} - 42 v + 25 u - 35 v - 35$

Step 4

Add $- 14 u$ and $25 u$ .

$10 u^{2} + 16 u v + 11 u - 42 v^{2} - 42 v - 35 v - 35$

Step 5

Subtract $35 v$ from $- 42 v$ .

$10 u^{2} + 16 u v + 11 u - 42 v^{2} - 77 v - 35$

Step 6

Move $11 u$ .

$10 u^{2} + 16 u v - 42 v^{2} + 11 u - 77 v - 35$