Basic Math Examples

(4 u + 7 v) (3 u - 6 v - 5)

$(4 u + 7 v) (3 u - 6 v - 5)$

Step 1

Expand

(4 u + 7 v) (3 u - 6 v - 5)

$(4 u + 7 v) (3 u - 6 v - 5)$ by multiplying each term in the first expression by each term in the second expression.

4 u (3 u) + 4 u (- 6 v) + 4 u \cdot - 5 + 7 v (3 u) + 7 v (- 6 v) + 7 v \cdot - 5

$4 u (3 u) + 4 u (- 6 v) + 4 u \cdot - 5 + 7 v (3 u) + 7 v (- 6 v) + 7 v \cdot - 5$

Step 2

Simplify each term.

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Step 2.1

Rewrite using the commutative property of multiplication.

4 \cdot 3 u \cdot u + 4 u (- 6 v) + 4 u \cdot - 5 + 7 v (3 u) + 7 v (- 6 v) + 7 v \cdot - 5

$4 \cdot 3 u \cdot u + 4 u (- 6 v) + 4 u \cdot - 5 + 7 v (3 u) + 7 v (- 6 v) + 7 v \cdot - 5$

Step 2.2

Multiply

u

$u$ by

u

$u$ by adding the exponents.

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Step 2.2.1

Move

u

$u$ .

4 \cdot 3 (u \cdot u) + 4 u (- 6 v) + 4 u \cdot - 5 + 7 v (3 u) + 7 v (- 6 v) + 7 v \cdot - 5

$4 \cdot 3 (u \cdot u) + 4 u (- 6 v) + 4 u \cdot - 5 + 7 v (3 u) + 7 v (- 6 v) + 7 v \cdot - 5$

Step 2.2.2

Multiply

u

$u$ by

u

$u$ .

4 \cdot 3 u^{2} + 4 u (- 6 v) + 4 u \cdot - 5 + 7 v (3 u) + 7 v (- 6 v) + 7 v \cdot - 5

$4 \cdot 3 u^{2} + 4 u (- 6 v) + 4 u \cdot - 5 + 7 v (3 u) + 7 v (- 6 v) + 7 v \cdot - 5$

4 \cdot 3 u^{2} + 4 u (- 6 v) + 4 u \cdot - 5 + 7 v (3 u) + 7 v (- 6 v) + 7 v \cdot - 5

$4 \cdot 3 u^{2} + 4 u (- 6 v) + 4 u \cdot - 5 + 7 v (3 u) + 7 v (- 6 v) + 7 v \cdot - 5$

Step 2.3

Multiply

4

$4$ by

3

$3$ .

12 u^{2} + 4 u (- 6 v) + 4 u \cdot - 5 + 7 v (3 u) + 7 v (- 6 v) + 7 v \cdot - 5

$12 u^{2} + 4 u (- 6 v) + 4 u \cdot - 5 + 7 v (3 u) + 7 v (- 6 v) + 7 v \cdot - 5$

Step 2.4

Rewrite using the commutative property of multiplication.

12 u^{2} + 4 \cdot - 6 u v + 4 u \cdot - 5 + 7 v (3 u) + 7 v (- 6 v) + 7 v \cdot - 5

$12 u^{2} + 4 \cdot - 6 u v + 4 u \cdot - 5 + 7 v (3 u) + 7 v (- 6 v) + 7 v \cdot - 5$

Step 2.5

Multiply

4

$4$ by

- 6

$- 6$ .

12 u^{2} - 24 u v + 4 u \cdot - 5 + 7 v (3 u) + 7 v (- 6 v) + 7 v \cdot - 5

$12 u^{2} - 24 u v + 4 u \cdot - 5 + 7 v (3 u) + 7 v (- 6 v) + 7 v \cdot - 5$

Step 2.6

Multiply

- 5

$- 5$ by

4

$4$ .

12 u^{2} - 24 u v - 20 u + 7 v (3 u) + 7 v (- 6 v) + 7 v \cdot - 5

$12 u^{2} - 24 u v - 20 u + 7 v (3 u) + 7 v (- 6 v) + 7 v \cdot - 5$

Step 2.7

Rewrite using the commutative property of multiplication.

12 u^{2} - 24 u v - 20 u + 7 \cdot 3 v u + 7 v (- 6 v) + 7 v \cdot - 5

$12 u^{2} - 24 u v - 20 u + 7 \cdot 3 v u + 7 v (- 6 v) + 7 v \cdot - 5$

Step 2.8

Multiply

7

$7$ by

3

$3$ .

12 u^{2} - 24 u v - 20 u + 21 v u + 7 v (- 6 v) + 7 v \cdot - 5

$12 u^{2} - 24 u v - 20 u + 21 v u + 7 v (- 6 v) + 7 v \cdot - 5$

Step 2.9

Rewrite using the commutative property of multiplication.

12 u^{2} - 24 u v - 20 u + 21 v u + 7 \cdot - 6 v \cdot v + 7 v \cdot - 5

$12 u^{2} - 24 u v - 20 u + 21 v u + 7 \cdot - 6 v \cdot v + 7 v \cdot - 5$

Step 2.10

Multiply

v

$v$ by

v

$v$ by adding the exponents.

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Step 2.10.1

Move

v

$v$ .

12 u^{2} - 24 u v - 20 u + 21 v u + 7 \cdot - 6 (v \cdot v) + 7 v \cdot - 5

$12 u^{2} - 24 u v - 20 u + 21 v u + 7 \cdot - 6 (v \cdot v) + 7 v \cdot - 5$

Step 2.10.2

Multiply

v

$v$ by

v

$v$ .

12 u^{2} - 24 u v - 20 u + 21 v u + 7 \cdot - 6 v^{2} + 7 v \cdot - 5

$12 u^{2} - 24 u v - 20 u + 21 v u + 7 \cdot - 6 v^{2} + 7 v \cdot - 5$

12 u^{2} - 24 u v - 20 u + 21 v u + 7 \cdot - 6 v^{2} + 7 v \cdot - 5

$12 u^{2} - 24 u v - 20 u + 21 v u + 7 \cdot - 6 v^{2} + 7 v \cdot - 5$

Step 2.11

Multiply

7

$7$ by

- 6

$- 6$ .

12 u^{2} - 24 u v - 20 u + 21 v u - 42 v^{2} + 7 v \cdot - 5

$12 u^{2} - 24 u v - 20 u + 21 v u - 42 v^{2} + 7 v \cdot - 5$

Step 2.12

Multiply

- 5

$- 5$ by

7

$7$ .

12 u^{2} - 24 u v - 20 u + 21 v u - 42 v^{2} - 35 v

$12 u^{2} - 24 u v - 20 u + 21 v u - 42 v^{2} - 35 v$

12 u^{2} - 24 u v - 20 u + 21 v u - 42 v^{2} - 35 v

$12 u^{2} - 24 u v - 20 u + 21 v u - 42 v^{2} - 35 v$

Step 3

Add

- 24 u v

$- 24 u v$ and

21 v u

$21 v u$ .

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Step 3.1

Move

v

$v$ .

12 u^{2} - 24 u v + 21 u v - 20 u - 42 v^{2} - 35 v

$12 u^{2} - 24 u v + 21 u v - 20 u - 42 v^{2} - 35 v$

Step 3.2

Add

$- 24 u v$ and

$21 u v$ .

$12 u^{2} - 3 u v - 20 u - 42 v^{2} - 35 v$

Step 4

Move

$- 20 u$ .

$12 u^{2} - 3 u v - 42 v^{2} - 20 u - 35 v$