Basic Math Examples

$(2 p^{4} - p^{3} + 7 p) (- 6 p^{2} - 3 p)$

Step 1

Expand $(2 p^{4} - p^{3} + 7 p) (- 6 p^{2} - 3 p)$ by multiplying each term in the first expression by each term in the second expression.

$2 p^{4} (- 6 p^{2}) + 2 p^{4} (- 3 p) - p^{3} (- 6 p^{2}) - p^{3} (- 3 p) + 7 p (- 6 p^{2}) + 7 p (- 3 p)$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite using the commutative property of multiplication.

$2 \cdot - 6 p^{4} p^{2} + 2 p^{4} (- 3 p) - p^{3} (- 6 p^{2}) - p^{3} (- 3 p) + 7 p (- 6 p^{2}) + 7 p (- 3 p)$

Step 2.1.2

Multiply $p^{4}$ by $p^{2}$ by adding the exponents.

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Step 2.1.2.1

Move $p^{2}$ .

$2 \cdot - 6 (p^{2} p^{4}) + 2 p^{4} (- 3 p) - p^{3} (- 6 p^{2}) - p^{3} (- 3 p) + 7 p (- 6 p^{2}) + 7 p (- 3 p)$

Step 2.1.2.2

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

$2 \cdot - 6 p^{2 + 4} + 2 p^{4} (- 3 p) - p^{3} (- 6 p^{2}) - p^{3} (- 3 p) + 7 p (- 6 p^{2}) + 7 p (- 3 p)$

Step 2.1.2.3

Add $2$ and $4$ .

$2 \cdot - 6 p^{6} + 2 p^{4} (- 3 p) - p^{3} (- 6 p^{2}) - p^{3} (- 3 p) + 7 p (- 6 p^{2}) + 7 p (- 3 p)$

Step 2.1.3

Multiply $2$ by $- 6$ .

$- 12 p^{6} + 2 p^{4} (- 3 p) - p^{3} (- 6 p^{2}) - p^{3} (- 3 p) + 7 p (- 6 p^{2}) + 7 p (- 3 p)$

Step 2.1.4

Rewrite using the commutative property of multiplication.

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

Step 2.1.5

Multiply $p^{4}$ by $p$ by adding the exponents.

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Step 2.1.5.1

Move $p$ .

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

Step 2.1.5.2

Multiply $p$ by $p^{4}$ .

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Step 2.1.5.2.1

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

$- 12 p^{6} + 2 \cdot - 3 (p^{1} p^{4}) - p^{3} (- 6 p^{2}) - p^{3} (- 3 p) + 7 p (- 6 p^{2}) + 7 p (- 3 p)$

Step 2.1.5.2.2

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

$- 12 p^{6} + 2 \cdot - 3 p^{1 + 4} - p^{3} (- 6 p^{2}) - p^{3} (- 3 p) + 7 p (- 6 p^{2}) + 7 p (- 3 p)$

Step 2.1.5.3

Add $1$ and $4$ .

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

Step 2.1.6

Multiply $2$ by $- 3$ .

$- 12 p^{6} - 6 p^{5} - p^{3} (- 6 p^{2}) - p^{3} (- 3 p) + 7 p (- 6 p^{2}) + 7 p (- 3 p)$

Step 2.1.7

Rewrite using the commutative property of multiplication.

$- 12 p^{6} - 6 p^{5} - 1 \cdot - 6 p^{3} p^{2} - p^{3} (- 3 p) + 7 p (- 6 p^{2}) + 7 p (- 3 p)$

Step 2.1.8

Multiply $p^{3}$ by $p^{2}$ by adding the exponents.

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Step 2.1.8.1

Move $p^{2}$ .

$- 12 p^{6} - 6 p^{5} - 1 \cdot - 6 (p^{2} p^{3}) - p^{3} (- 3 p) + 7 p (- 6 p^{2}) + 7 p (- 3 p)$

Step 2.1.8.2

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

$- 12 p^{6} - 6 p^{5} - 1 \cdot - 6 p^{2 + 3} - p^{3} (- 3 p) + 7 p (- 6 p^{2}) + 7 p (- 3 p)$

Step 2.1.8.3

Add $2$ and $3$ .

$- 12 p^{6} - 6 p^{5} - 1 \cdot - 6 p^{5} - p^{3} (- 3 p) + 7 p (- 6 p^{2}) + 7 p (- 3 p)$

Step 2.1.9

Multiply $- 1$ by $- 6$ .

$- 12 p^{6} - 6 p^{5} + 6 p^{5} - p^{3} (- 3 p) + 7 p (- 6 p^{2}) + 7 p (- 3 p)$

Step 2.1.10

Rewrite using the commutative property of multiplication.

$- 12 p^{6} - 6 p^{5} + 6 p^{5} - 1 \cdot - 3 p^{3} p + 7 p (- 6 p^{2}) + 7 p (- 3 p)$

Step 2.1.11

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

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Step 2.1.11.1

Move $p$ .

$- 12 p^{6} - 6 p^{5} + 6 p^{5} - 1 \cdot - 3 (p \cdot p^{3}) + 7 p (- 6 p^{2}) + 7 p (- 3 p)$

Step 2.1.11.2

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

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Step 2.1.11.2.1

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

$- 12 p^{6} - 6 p^{5} + 6 p^{5} - 1 \cdot - 3 (p^{1} p^{3}) + 7 p (- 6 p^{2}) + 7 p (- 3 p)$

Step 2.1.11.2.2

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

$- 12 p^{6} - 6 p^{5} + 6 p^{5} - 1 \cdot - 3 p^{1 + 3} + 7 p (- 6 p^{2}) + 7 p (- 3 p)$

Step 2.1.11.3

Add $1$ and $3$ .

$- 12 p^{6} - 6 p^{5} + 6 p^{5} - 1 \cdot - 3 p^{4} + 7 p (- 6 p^{2}) + 7 p (- 3 p)$

Step 2.1.12

Multiply $- 1$ by $- 3$ .

$- 12 p^{6} - 6 p^{5} + 6 p^{5} + 3 p^{4} + 7 p (- 6 p^{2}) + 7 p (- 3 p)$

Step 2.1.13

Rewrite using the commutative property of multiplication.

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

Step 2.1.14

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

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Step 2.1.14.1

Move $p^{2}$ .

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

Step 2.1.14.2

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

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Step 2.1.14.2.1

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

$- 12 p^{6} - 6 p^{5} + 6 p^{5} + 3 p^{4} + 7 \cdot - 6 (p^{2} p^{1}) + 7 p (- 3 p)$

Step 2.1.14.2.2

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

$- 12 p^{6} - 6 p^{5} + 6 p^{5} + 3 p^{4} + 7 \cdot - 6 p^{2 + 1} + 7 p (- 3 p)$

Step 2.1.14.3

Add $2$ and $1$ .

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

Step 2.1.15

Multiply $7$ by $- 6$ .

$- 12 p^{6} - 6 p^{5} + 6 p^{5} + 3 p^{4} - 42 p^{3} + 7 p (- 3 p)$

Step 2.1.16

Rewrite using the commutative property of multiplication.

$- 12 p^{6} - 6 p^{5} + 6 p^{5} + 3 p^{4} - 42 p^{3} + 7 \cdot - 3 p \cdot p$

Step 2.1.17

Multiply $p$ by $p$ by adding the exponents.

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Step 2.1.17.1

Move $p$ .

$- 12 p^{6} - 6 p^{5} + 6 p^{5} + 3 p^{4} - 42 p^{3} + 7 \cdot - 3 (p \cdot p)$

Step 2.1.17.2

Multiply $p$ by $p$ .

$- 12 p^{6} - 6 p^{5} + 6 p^{5} + 3 p^{4} - 42 p^{3} + 7 \cdot - 3 p^{2}$

Step 2.1.18

Multiply $7$ by $- 3$ .

$- 12 p^{6} - 6 p^{5} + 6 p^{5} + 3 p^{4} - 42 p^{3} - 21 p^{2}$

Step 2.2

Combine the opposite terms in $- 12 p^{6} - 6 p^{5} + 6 p^{5} + 3 p^{4} - 42 p^{3} - 21 p^{2}$ .

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Step 2.2.1

Add $- 6 p^{5}$ and $6 p^{5}$ .

$- 12 p^{6} + 0 + 3 p^{4} - 42 p^{3} - 21 p^{2}$

Step 2.2.2

Add $- 12 p^{6}$ and $0$ .

$- 12 p^{6} + 3 p^{4} - 42 p^{3} - 21 p^{2}$