Basic Math Examples

(- 6 w^{2} + 9 w - 5) + (7 w^{2} + 9 w - 7) (9 w^{2} + 5 w - 2)

$(- 6 w^{2} + 9 w - 5) + (7 w^{2} + 9 w - 7) (9 w^{2} + 5 w - 2)$

Step 1

Simplify each term.

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Step 1.1

Expand

$(7 w^{2} + 9 w - 7) (9 w^{2} + 5 w - 2)$ by multiplying each term in the first expression by each term in the second expression.

$- 6 w^{2} + 9 w - 5 + 7 w^{2} (9 w^{2}) + 7 w^{2} (5 w) + 7 w^{2} \cdot - 2 + 9 w (9 w^{2}) + 9 w (5 w) + 9 w \cdot - 2 - 7 (9 w^{2}) - 7 (5 w) - 7 \cdot - 2$

Step 1.2

Simplify each term.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

$- 6 w^{2} + 9 w - 5 + 7 \cdot 9 w^{2} w^{2} + 7 w^{2} (5 w) + 7 w^{2} \cdot - 2 + 9 w (9 w^{2}) + 9 w (5 w) + 9 w \cdot - 2 - 7 (9 w^{2}) - 7 (5 w) - 7 \cdot - 2$

Step 1.2.2

Multiply

$w^{2}$ by

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

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Step 1.2.2.1

Move

$w^{2}$ .

$- 6 w^{2} + 9 w - 5 + 7 \cdot 9 (w^{2} w^{2}) + 7 w^{2} (5 w) + 7 w^{2} \cdot - 2 + 9 w (9 w^{2}) + 9 w (5 w) + 9 w \cdot - 2 - 7 (9 w^{2}) - 7 (5 w) - 7 \cdot - 2$

Step 1.2.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 6 w^{2} + 9 w - 5 + 7 \cdot 9 w^{2 + 2} + 7 w^{2} (5 w) + 7 w^{2} \cdot - 2 + 9 w (9 w^{2}) + 9 w (5 w) + 9 w \cdot - 2 - 7 (9 w^{2}) - 7 (5 w) - 7 \cdot - 2$

Step 1.2.2.3

Add

$2$ and

$2$ .

$- 6 w^{2} + 9 w - 5 + 7 \cdot 9 w^{4} + 7 w^{2} (5 w) + 7 w^{2} \cdot - 2 + 9 w (9 w^{2}) + 9 w (5 w) + 9 w \cdot - 2 - 7 (9 w^{2}) - 7 (5 w) - 7 \cdot - 2$

Step 1.2.3

Multiply

$7$ by

$9$ .

$- 6 w^{2} + 9 w - 5 + 63 w^{4} + 7 w^{2} (5 w) + 7 w^{2} \cdot - 2 + 9 w (9 w^{2}) + 9 w (5 w) + 9 w \cdot - 2 - 7 (9 w^{2}) - 7 (5 w) - 7 \cdot - 2$

Step 1.2.4

Rewrite using the commutative property of multiplication.

$- 6 w^{2} + 9 w - 5 + 63 w^{4} + 7 \cdot 5 w^{2} w + 7 w^{2} \cdot - 2 + 9 w (9 w^{2}) + 9 w (5 w) + 9 w \cdot - 2 - 7 (9 w^{2}) - 7 (5 w) - 7 \cdot - 2$

Step 1.2.5

Multiply

$w^{2}$ by

$w$ by adding the exponents.

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Step 1.2.5.1

Move

$w$ .

$- 6 w^{2} + 9 w - 5 + 63 w^{4} + 7 \cdot 5 (w \cdot w^{2}) + 7 w^{2} \cdot - 2 + 9 w (9 w^{2}) + 9 w (5 w) + 9 w \cdot - 2 - 7 (9 w^{2}) - 7 (5 w) - 7 \cdot - 2$

Step 1.2.5.2

Multiply

$w$ by

$w^{2}$ .

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Step 1.2.5.2.1

Raise

$w$ to the power of

$1$ .

$- 6 w^{2} + 9 w - 5 + 63 w^{4} + 7 \cdot 5 (w^{1} w^{2}) + 7 w^{2} \cdot - 2 + 9 w (9 w^{2}) + 9 w (5 w) + 9 w \cdot - 2 - 7 (9 w^{2}) - 7 (5 w) - 7 \cdot - 2$

Step 1.2.5.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 6 w^{2} + 9 w - 5 + 63 w^{4} + 7 \cdot 5 w^{1 + 2} + 7 w^{2} \cdot - 2 + 9 w (9 w^{2}) + 9 w (5 w) + 9 w \cdot - 2 - 7 (9 w^{2}) - 7 (5 w) - 7 \cdot - 2$

Step 1.2.5.3

Add

$1$ and

$2$ .

$- 6 w^{2} + 9 w - 5 + 63 w^{4} + 7 \cdot 5 w^{3} + 7 w^{2} \cdot - 2 + 9 w (9 w^{2}) + 9 w (5 w) + 9 w \cdot - 2 - 7 (9 w^{2}) - 7 (5 w) - 7 \cdot - 2$

Step 1.2.6

Multiply

$7$ by

$5$ .

$- 6 w^{2} + 9 w - 5 + 63 w^{4} + 35 w^{3} + 7 w^{2} \cdot - 2 + 9 w (9 w^{2}) + 9 w (5 w) + 9 w \cdot - 2 - 7 (9 w^{2}) - 7 (5 w) - 7 \cdot - 2$

Step 1.2.7

Multiply

$- 2$ by

$7$ .

$- 6 w^{2} + 9 w - 5 + 63 w^{4} + 35 w^{3} - 14 w^{2} + 9 w (9 w^{2}) + 9 w (5 w) + 9 w \cdot - 2 - 7 (9 w^{2}) - 7 (5 w) - 7 \cdot - 2$

Step 1.2.8

Rewrite using the commutative property of multiplication.

$- 6 w^{2} + 9 w - 5 + 63 w^{4} + 35 w^{3} - 14 w^{2} + 9 \cdot 9 w \cdot w^{2} + 9 w (5 w) + 9 w \cdot - 2 - 7 (9 w^{2}) - 7 (5 w) - 7 \cdot - 2$

Step 1.2.9

Multiply

$w$ by

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

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Step 1.2.9.1

Move

$w^{2}$ .

$- 6 w^{2} + 9 w - 5 + 63 w^{4} + 35 w^{3} - 14 w^{2} + 9 \cdot 9 (w^{2} w) + 9 w (5 w) + 9 w \cdot - 2 - 7 (9 w^{2}) - 7 (5 w) - 7 \cdot - 2$

Step 1.2.9.2

Multiply

$w^{2}$ by

$w$ .

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Step 1.2.9.2.1

Raise

$w$ to the power of

$1$ .

$- 6 w^{2} + 9 w - 5 + 63 w^{4} + 35 w^{3} - 14 w^{2} + 9 \cdot 9 (w^{2} w^{1}) + 9 w (5 w) + 9 w \cdot - 2 - 7 (9 w^{2}) - 7 (5 w) - 7 \cdot - 2$

Step 1.2.9.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 6 w^{2} + 9 w - 5 + 63 w^{4} + 35 w^{3} - 14 w^{2} + 9 \cdot 9 w^{2 + 1} + 9 w (5 w) + 9 w \cdot - 2 - 7 (9 w^{2}) - 7 (5 w) - 7 \cdot - 2$

Step 1.2.9.3

Add

$2$ and

$1$ .

$- 6 w^{2} + 9 w - 5 + 63 w^{4} + 35 w^{3} - 14 w^{2} + 9 \cdot 9 w^{3} + 9 w (5 w) + 9 w \cdot - 2 - 7 (9 w^{2}) - 7 (5 w) - 7 \cdot - 2$

Step 1.2.10

Multiply

$9$ by

$9$ .

$- 6 w^{2} + 9 w - 5 + 63 w^{4} + 35 w^{3} - 14 w^{2} + 81 w^{3} + 9 w (5 w) + 9 w \cdot - 2 - 7 (9 w^{2}) - 7 (5 w) - 7 \cdot - 2$

Step 1.2.11

Rewrite using the commutative property of multiplication.

$- 6 w^{2} + 9 w - 5 + 63 w^{4} + 35 w^{3} - 14 w^{2} + 81 w^{3} + 9 \cdot 5 w \cdot w + 9 w \cdot - 2 - 7 (9 w^{2}) - 7 (5 w) - 7 \cdot - 2$

Step 1.2.12

Multiply

$w$ by

$w$ by adding the exponents.

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Step 1.2.12.1

Move

$w$ .

$- 6 w^{2} + 9 w - 5 + 63 w^{4} + 35 w^{3} - 14 w^{2} + 81 w^{3} + 9 \cdot 5 (w \cdot w) + 9 w \cdot - 2 - 7 (9 w^{2}) - 7 (5 w) - 7 \cdot - 2$

Step 1.2.12.2

Multiply

$w$ by

$w$ .

$- 6 w^{2} + 9 w - 5 + 63 w^{4} + 35 w^{3} - 14 w^{2} + 81 w^{3} + 9 \cdot 5 w^{2} + 9 w \cdot - 2 - 7 (9 w^{2}) - 7 (5 w) - 7 \cdot - 2$

Step 1.2.13

Multiply

$9$ by

$5$ .

$- 6 w^{2} + 9 w - 5 + 63 w^{4} + 35 w^{3} - 14 w^{2} + 81 w^{3} + 45 w^{2} + 9 w \cdot - 2 - 7 (9 w^{2}) - 7 (5 w) - 7 \cdot - 2$

Step 1.2.14

Multiply

$- 2$ by

$9$ .

$- 6 w^{2} + 9 w - 5 + 63 w^{4} + 35 w^{3} - 14 w^{2} + 81 w^{3} + 45 w^{2} - 18 w - 7 (9 w^{2}) - 7 (5 w) - 7 \cdot - 2$

Step 1.2.15

Multiply

$9$ by

$- 7$ .

$- 6 w^{2} + 9 w - 5 + 63 w^{4} + 35 w^{3} - 14 w^{2} + 81 w^{3} + 45 w^{2} - 18 w - 63 w^{2} - 7 (5 w) - 7 \cdot - 2$

Step 1.2.16

Multiply

$5$ by

$- 7$ .

$- 6 w^{2} + 9 w - 5 + 63 w^{4} + 35 w^{3} - 14 w^{2} + 81 w^{3} + 45 w^{2} - 18 w - 63 w^{2} - 35 w - 7 \cdot - 2$

Step 1.2.17

Multiply

$- 7$ by

$- 2$ .

$- 6 w^{2} + 9 w - 5 + 63 w^{4} + 35 w^{3} - 14 w^{2} + 81 w^{3} + 45 w^{2} - 18 w - 63 w^{2} - 35 w + 14$

Step 1.3

Add

$35 w^{3}$ and

$81 w^{3}$ .

$- 6 w^{2} + 9 w - 5 + 63 w^{4} + 116 w^{3} - 14 w^{2} + 45 w^{2} - 18 w - 63 w^{2} - 35 w + 14$

Step 1.4

Add

$- 14 w^{2}$ and

$45 w^{2}$ .

$- 6 w^{2} + 9 w - 5 + 63 w^{4} + 116 w^{3} + 31 w^{2} - 18 w - 63 w^{2} - 35 w + 14$

Step 1.5

Subtract

$63 w^{2}$ from

$31 w^{2}$ .

$- 6 w^{2} + 9 w - 5 + 63 w^{4} + 116 w^{3} - 32 w^{2} - 18 w - 35 w + 14$

Step 1.6

Subtract

$35 w$ from

$- 18 w$ .

$- 6 w^{2} + 9 w - 5 + 63 w^{4} + 116 w^{3} - 32 w^{2} - 53 w + 14$

Step 2

Simplify by adding terms.

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Step 2.1

Subtract

$32 w^{2}$ from

$- 6 w^{2}$ .

$- 38 w^{2} + 9 w - 5 + 63 w^{4} + 116 w^{3} - 53 w + 14$

Step 2.2

Subtract

$53 w$ from

$9 w$ .

$- 38 w^{2} - 5 + 63 w^{4} + 116 w^{3} - 44 w + 14$

Step 2.3

Simplify the expression.

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Step 2.3.1

Add

$- 5$ and

$14$ .

$- 38 w^{2} + 63 w^{4} + 116 w^{3} - 44 w + 9$

Step 2.3.2

Move

$- 38 w^{2}$ .

$63 w^{4} + 116 w^{3} - 38 w^{2} - 44 w + 9$