Algebra Examples

$- 5 q^{2} w^{3} (4 q + 7 w) + 4 q w^{2} (7 q^{2} w + 2 q) - 3 q w (3 q^{2} w^{2} + 9)$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$- 5 q^{2} w^{3} (4 q) - 5 q^{2} w^{3} (7 w) + 4 q w^{2} (7 q^{2} w + 2 q) - 3 q w (3 q^{2} w^{2} + 9)$

Step 1.2

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

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Step 1.2.1

Move $q$ .

$- 5 (q \cdot q^{2}) w^{3} \cdot 4 - 5 q^{2} w^{3} (7 w) + 4 q w^{2} (7 q^{2} w + 2 q) - 3 q w (3 q^{2} w^{2} + 9)$

Step 1.2.2

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

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Step 1.2.2.1

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

$- 5 (q^{1} q^{2}) w^{3} \cdot 4 - 5 q^{2} w^{3} (7 w) + 4 q w^{2} (7 q^{2} w + 2 q) - 3 q w (3 q^{2} w^{2} + 9)$

Step 1.2.2.2

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

$- 5 q^{1 + 2} w^{3} \cdot 4 - 5 q^{2} w^{3} (7 w) + 4 q w^{2} (7 q^{2} w + 2 q) - 3 q w (3 q^{2} w^{2} + 9)$

Step 1.2.3

Add $1$ and $2$ .

$- 5 q^{3} w^{3} \cdot 4 - 5 q^{2} w^{3} (7 w) + 4 q w^{2} (7 q^{2} w + 2 q) - 3 q w (3 q^{2} w^{2} + 9)$

Step 1.3

Multiply $w^{3}$ by $w$ by adding the exponents.

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Step 1.3.1

Move $w$ .

$- 5 q^{3} w^{3} \cdot 4 - 5 q^{2} (w \cdot w^{3}) \cdot 7 + 4 q w^{2} (7 q^{2} w + 2 q) - 3 q w (3 q^{2} w^{2} + 9)$

Step 1.3.2

Multiply $w$ by $w^{3}$ .

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Step 1.3.2.1

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

$- 5 q^{3} w^{3} \cdot 4 - 5 q^{2} (w^{1} w^{3}) \cdot 7 + 4 q w^{2} (7 q^{2} w + 2 q) - 3 q w (3 q^{2} w^{2} + 9)$

Step 1.3.2.2

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

$- 5 q^{3} w^{3} \cdot 4 - 5 q^{2} w^{1 + 3} \cdot 7 + 4 q w^{2} (7 q^{2} w + 2 q) - 3 q w (3 q^{2} w^{2} + 9)$

Step 1.3.3

Add $1$ and $3$ .

$- 5 q^{3} w^{3} \cdot 4 - 5 q^{2} w^{4} \cdot 7 + 4 q w^{2} (7 q^{2} w + 2 q) - 3 q w (3 q^{2} w^{2} + 9)$

Step 1.4

Simplify each term.

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Step 1.4.1

Multiply $4$ by $- 5$ .

$- 20 q^{3} w^{3} - 5 q^{2} w^{4} \cdot 7 + 4 q w^{2} (7 q^{2} w + 2 q) - 3 q w (3 q^{2} w^{2} + 9)$

Step 1.4.2

Multiply $7$ by $- 5$ .

$- 20 q^{3} w^{3} - 35 q^{2} w^{4} + 4 q w^{2} (7 q^{2} w + 2 q) - 3 q w (3 q^{2} w^{2} + 9)$

Step 1.5

Apply the distributive property.

$- 20 q^{3} w^{3} - 35 q^{2} w^{4} + 4 q w^{2} (7 q^{2} w) + 4 q w^{2} (2 q) - 3 q w (3 q^{2} w^{2} + 9)$

Step 1.6

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

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Step 1.6.1

Move $q^{2}$ .

$- 20 q^{3} w^{3} - 35 q^{2} w^{4} + 4 (q^{2} q) w^{2} (7 w) + 4 q w^{2} (2 q) - 3 q w (3 q^{2} w^{2} + 9)$

Step 1.6.2

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

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Step 1.6.2.1

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

$- 20 q^{3} w^{3} - 35 q^{2} w^{4} + 4 (q^{2} q^{1}) w^{2} (7 w) + 4 q w^{2} (2 q) - 3 q w (3 q^{2} w^{2} + 9)$

Step 1.6.2.2

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

$- 20 q^{3} w^{3} - 35 q^{2} w^{4} + 4 q^{2 + 1} w^{2} (7 w) + 4 q w^{2} (2 q) - 3 q w (3 q^{2} w^{2} + 9)$

Step 1.6.3

Add $2$ and $1$ .

$- 20 q^{3} w^{3} - 35 q^{2} w^{4} + 4 q^{3} w^{2} (7 w) + 4 q w^{2} (2 q) - 3 q w (3 q^{2} w^{2} + 9)$

Step 1.7

Multiply $q$ by $q$ by adding the exponents.

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Step 1.7.1

Move $q$ .

$- 20 q^{3} w^{3} - 35 q^{2} w^{4} + 4 q^{3} w^{2} (7 w) + 4 (q \cdot q) w^{2} \cdot 2 - 3 q w (3 q^{2} w^{2} + 9)$

Step 1.7.2

Multiply $q$ by $q$ .

$- 20 q^{3} w^{3} - 35 q^{2} w^{4} + 4 q^{3} w^{2} (7 w) + 4 q^{2} w^{2} \cdot 2 - 3 q w (3 q^{2} w^{2} + 9)$

Step 1.8

Simplify each term.

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Step 1.8.1

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

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Step 1.8.1.1

Move $w$ .

$- 20 q^{3} w^{3} - 35 q^{2} w^{4} + 4 q^{3} (w \cdot w^{2}) \cdot 7 + 4 q^{2} w^{2} \cdot 2 - 3 q w (3 q^{2} w^{2} + 9)$

Step 1.8.1.2

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

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Step 1.8.1.2.1

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

$- 20 q^{3} w^{3} - 35 q^{2} w^{4} + 4 q^{3} (w^{1} w^{2}) \cdot 7 + 4 q^{2} w^{2} \cdot 2 - 3 q w (3 q^{2} w^{2} + 9)$

Step 1.8.1.2.2

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

$- 20 q^{3} w^{3} - 35 q^{2} w^{4} + 4 q^{3} w^{1 + 2} \cdot 7 + 4 q^{2} w^{2} \cdot 2 - 3 q w (3 q^{2} w^{2} + 9)$

Step 1.8.1.3

Add $1$ and $2$ .

$- 20 q^{3} w^{3} - 35 q^{2} w^{4} + 4 q^{3} w^{3} \cdot 7 + 4 q^{2} w^{2} \cdot 2 - 3 q w (3 q^{2} w^{2} + 9)$

Step 1.8.2

Multiply $7$ by $4$ .

$- 20 q^{3} w^{3} - 35 q^{2} w^{4} + 28 q^{3} w^{3} + 4 q^{2} w^{2} \cdot 2 - 3 q w (3 q^{2} w^{2} + 9)$

Step 1.8.3

Multiply $2$ by $4$ .

$- 20 q^{3} w^{3} - 35 q^{2} w^{4} + 28 q^{3} w^{3} + 8 q^{2} w^{2} - 3 q w (3 q^{2} w^{2} + 9)$

Step 1.9

Apply the distributive property.

$- 20 q^{3} w^{3} - 35 q^{2} w^{4} + 28 q^{3} w^{3} + 8 q^{2} w^{2} - 3 q w (3 q^{2} w^{2}) - 3 q w \cdot 9$

Step 1.10

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

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Step 1.10.1

Move $q^{2}$ .

$- 20 q^{3} w^{3} - 35 q^{2} w^{4} + 28 q^{3} w^{3} + 8 q^{2} w^{2} - 3 (q^{2} q) w (3 w^{2}) - 3 q w \cdot 9$

Step 1.10.2

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

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Step 1.10.2.1

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

$- 20 q^{3} w^{3} - 35 q^{2} w^{4} + 28 q^{3} w^{3} + 8 q^{2} w^{2} - 3 (q^{2} q^{1}) w (3 w^{2}) - 3 q w \cdot 9$

Step 1.10.2.2

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

$- 20 q^{3} w^{3} - 35 q^{2} w^{4} + 28 q^{3} w^{3} + 8 q^{2} w^{2} - 3 q^{2 + 1} w (3 w^{2}) - 3 q w \cdot 9$

Step 1.10.3

Add $2$ and $1$ .

$- 20 q^{3} w^{3} - 35 q^{2} w^{4} + 28 q^{3} w^{3} + 8 q^{2} w^{2} - 3 q^{3} w (3 w^{2}) - 3 q w \cdot 9$

Step 1.11

Multiply $9$ by $- 3$ .

$- 20 q^{3} w^{3} - 35 q^{2} w^{4} + 28 q^{3} w^{3} + 8 q^{2} w^{2} - 3 q^{3} w (3 w^{2}) - 27 q w$

Step 1.12

Simplify each term.

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Step 1.12.1

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

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Step 1.12.1.1

Move $w^{2}$ .

$- 20 q^{3} w^{3} - 35 q^{2} w^{4} + 28 q^{3} w^{3} + 8 q^{2} w^{2} - 3 q^{3} (w^{2} w) \cdot 3 - 27 q w$

Step 1.12.1.2

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

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Step 1.12.1.2.1

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

$- 20 q^{3} w^{3} - 35 q^{2} w^{4} + 28 q^{3} w^{3} + 8 q^{2} w^{2} - 3 q^{3} (w^{2} w^{1}) \cdot 3 - 27 q w$

Step 1.12.1.2.2

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

$- 20 q^{3} w^{3} - 35 q^{2} w^{4} + 28 q^{3} w^{3} + 8 q^{2} w^{2} - 3 q^{3} w^{2 + 1} \cdot 3 - 27 q w$

Step 1.12.1.3

Add $2$ and $1$ .

$- 20 q^{3} w^{3} - 35 q^{2} w^{4} + 28 q^{3} w^{3} + 8 q^{2} w^{2} - 3 q^{3} w^{3} \cdot 3 - 27 q w$

Step 1.12.2

Multiply $3$ by $- 3$ .

$- 20 q^{3} w^{3} - 35 q^{2} w^{4} + 28 q^{3} w^{3} + 8 q^{2} w^{2} - 9 q^{3} w^{3} - 27 q w$

Step 2

Simplify by adding terms.

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Step 2.1

Add $- 20 q^{3} w^{3}$ and $28 q^{3} w^{3}$ .

$8 q^{3} w^{3} - 35 q^{2} w^{4} + 8 q^{2} w^{2} - 9 q^{3} w^{3} - 27 q w$

Step 2.2

Subtract $9 q^{3} w^{3}$ from $8 q^{3} w^{3}$ .

$- q^{3} w^{3} - 35 q^{2} w^{4} + 8 q^{2} w^{2} - 27 q w$