Algebra Examples

$(9 v^{4} + 2) + v^{2} (v^{2} w^{2} + 2 w^{3} - 2 v^{2}) - (- 13 v^{2} w^{3} + 7 v^{4})$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$9 v^{4} + 2 + v^{2} (v^{2} w^{2}) + v^{2} (2 w^{3}) + v^{2} (- 2 v^{2}) - (- 13 v^{2} w^{3} + 7 v^{4})$

Step 1.2

Simplify.

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Step 1.2.1

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

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Step 1.2.1.1

Move $v^{2}$ .

$9 v^{4} + 2 + v^{2} v^{2} w^{2} + v^{2} (2 w^{3}) + v^{2} (- 2 v^{2}) - (- 13 v^{2} w^{3} + 7 v^{4})$

Step 1.2.1.2

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

$9 v^{4} + 2 + v^{2 + 2} w^{2} + v^{2} (2 w^{3}) + v^{2} (- 2 v^{2}) - (- 13 v^{2} w^{3} + 7 v^{4})$

Step 1.2.1.3

Add $2$ and $2$ .

$9 v^{4} + 2 + v^{4} w^{2} + v^{2} (2 w^{3}) + v^{2} (- 2 v^{2}) - (- 13 v^{2} w^{3} + 7 v^{4})$

Step 1.2.2

Rewrite using the commutative property of multiplication.

$9 v^{4} + 2 + v^{4} w^{2} + 2 v^{2} w^{3} + v^{2} (- 2 v^{2}) - (- 13 v^{2} w^{3} + 7 v^{4})$

Step 1.2.3

Rewrite using the commutative property of multiplication.

$9 v^{4} + 2 + v^{4} w^{2} + 2 v^{2} w^{3} - 2 v^{2} v^{2} - (- 13 v^{2} w^{3} + 7 v^{4})$

Step 1.3

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

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Step 1.3.1

Move $v^{2}$ .

$9 v^{4} + 2 + v^{4} w^{2} + 2 v^{2} w^{3} - 2 (v^{2} v^{2}) - (- 13 v^{2} w^{3} + 7 v^{4})$

Step 1.3.2

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

$9 v^{4} + 2 + v^{4} w^{2} + 2 v^{2} w^{3} - 2 v^{2 + 2} - (- 13 v^{2} w^{3} + 7 v^{4})$

Step 1.3.3

Add $2$ and $2$ .

$9 v^{4} + 2 + v^{4} w^{2} + 2 v^{2} w^{3} - 2 v^{4} - (- 13 v^{2} w^{3} + 7 v^{4})$

Step 1.4

Apply the distributive property.

$9 v^{4} + 2 + v^{4} w^{2} + 2 v^{2} w^{3} - 2 v^{4} - (- 13 v^{2} w^{3}) - (7 v^{4})$

Step 1.5

Multiply $- 13$ by $- 1$ .

$9 v^{4} + 2 + v^{4} w^{2} + 2 v^{2} w^{3} - 2 v^{4} + 13 (v^{2} w^{3}) - (7 v^{4})$

Step 1.6

Multiply $7$ by $- 1$ .

$9 v^{4} + 2 + v^{4} w^{2} + 2 v^{2} w^{3} - 2 v^{4} + 13 v^{2} w^{3} - 7 v^{4}$

Step 2

Simplify by adding terms.

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Step 2.1

Subtract $2 v^{4}$ from $9 v^{4}$ .

$7 v^{4} + 2 + v^{4} w^{2} + 2 v^{2} w^{3} + 13 v^{2} w^{3} - 7 v^{4}$

Step 2.2

Combine the opposite terms in $7 v^{4} + 2 + v^{4} w^{2} + 2 v^{2} w^{3} + 13 v^{2} w^{3} - 7 v^{4}$ .

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Step 2.2.1

Subtract $7 v^{4}$ from $7 v^{4}$ .

$2 + v^{4} w^{2} + 2 v^{2} w^{3} + 13 v^{2} w^{3} + 0$

Step 2.2.2

Add $2 + v^{4} w^{2} + 2 v^{2} w^{3} + 13 v^{2} w^{3}$ and $0$ .

$2 + v^{4} w^{2} + 2 v^{2} w^{3} + 13 v^{2} w^{3}$

Step 2.3

Add $2 v^{2} w^{3}$ and $13 v^{2} w^{3}$ .

$2 + v^{4} w^{2} + 15 v^{2} w^{3}$

Step 2.4

Move $2$ .

$v^{4} w^{2} + 15 v^{2} w^{3} + 2$