Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Simplify.

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Step 1.2.1

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

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Step 1.2.1.1

Move $n^{3}$ .

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

Step 1.2.1.2

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

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Step 1.2.1.2.1

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

$4 (n^{3} n^{1}) (2 p^{2}) + 4 n (- 3 n p^{2}) + 4 n (5 n) + 4 p (6 n^{2} p - 2 n p^{2} + 3 p)$

Step 1.2.1.2.2

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

$4 n^{3 + 1} (2 p^{2}) + 4 n (- 3 n p^{2}) + 4 n (5 n) + 4 p (6 n^{2} p - 2 n p^{2} + 3 p)$

Step 1.2.1.3

Add $3$ and $1$ .

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

Step 1.2.2

Multiply $n$ by $n$ by adding the exponents.

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Step 1.2.2.1

Move $n$ .

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

Step 1.2.2.2

Multiply $n$ by $n$ .

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

Step 1.2.3

Rewrite using the commutative property of multiplication.

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

Step 1.3

Simplify each term.

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Step 1.3.1

Rewrite using the commutative property of multiplication.

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

Step 1.3.2

Multiply $4$ by $2$ .

$8 n^{4} p^{2} + 4 n^{2} (- 3 p^{2}) + 4 \cdot 5 n \cdot n + 4 p (6 n^{2} p - 2 n p^{2} + 3 p)$

Step 1.3.3

Rewrite using the commutative property of multiplication.

$8 n^{4} p^{2} + 4 \cdot - 3 n^{2} p^{2} + 4 \cdot 5 n \cdot n + 4 p (6 n^{2} p - 2 n p^{2} + 3 p)$

Step 1.3.4

Multiply $4$ by $- 3$ .

$8 n^{4} p^{2} - 12 n^{2} p^{2} + 4 \cdot 5 n \cdot n + 4 p (6 n^{2} p - 2 n p^{2} + 3 p)$

Step 1.3.5

Multiply $n$ by $n$ by adding the exponents.

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Step 1.3.5.1

Move $n$ .

$8 n^{4} p^{2} - 12 n^{2} p^{2} + 4 \cdot 5 (n \cdot n) + 4 p (6 n^{2} p - 2 n p^{2} + 3 p)$

Step 1.3.5.2

Multiply $n$ by $n$ .

$8 n^{4} p^{2} - 12 n^{2} p^{2} + 4 \cdot 5 n^{2} + 4 p (6 n^{2} p - 2 n p^{2} + 3 p)$

Step 1.3.6

Multiply $4$ by $5$ .

$8 n^{4} p^{2} - 12 n^{2} p^{2} + 20 n^{2} + 4 p (6 n^{2} p - 2 n p^{2} + 3 p)$

Step 1.4

Apply the distributive property.

$8 n^{4} p^{2} - 12 n^{2} p^{2} + 20 n^{2} + 4 p (6 n^{2} p) + 4 p (- 2 n p^{2}) + 4 p (3 p)$

Step 1.5

Simplify.

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Step 1.5.1

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

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Step 1.5.1.1

Move $p$ .

$8 n^{4} p^{2} - 12 n^{2} p^{2} + 20 n^{2} + 4 (p \cdot p) (6 n^{2}) + 4 p (- 2 n p^{2}) + 4 p (3 p)$

Step 1.5.1.2

Multiply $p$ by $p$ .

$8 n^{4} p^{2} - 12 n^{2} p^{2} + 20 n^{2} + 4 p^{2} (6 n^{2}) + 4 p (- 2 n p^{2}) + 4 p (3 p)$

Step 1.5.2

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

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Step 1.5.2.1

Move $p^{2}$ .

$8 n^{4} p^{2} - 12 n^{2} p^{2} + 20 n^{2} + 4 p^{2} (6 n^{2}) + 4 (p^{2} p) (- 2 n) + 4 p (3 p)$

Step 1.5.2.2

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

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Step 1.5.2.2.1

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

$8 n^{4} p^{2} - 12 n^{2} p^{2} + 20 n^{2} + 4 p^{2} (6 n^{2}) + 4 (p^{2} p^{1}) (- 2 n) + 4 p (3 p)$

Step 1.5.2.2.2

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

$8 n^{4} p^{2} - 12 n^{2} p^{2} + 20 n^{2} + 4 p^{2} (6 n^{2}) + 4 p^{2 + 1} (- 2 n) + 4 p (3 p)$

Step 1.5.2.3

Add $2$ and $1$ .

$8 n^{4} p^{2} - 12 n^{2} p^{2} + 20 n^{2} + 4 p^{2} (6 n^{2}) + 4 p^{3} (- 2 n) + 4 p (3 p)$

Step 1.5.3

Rewrite using the commutative property of multiplication.

$8 n^{4} p^{2} - 12 n^{2} p^{2} + 20 n^{2} + 4 p^{2} (6 n^{2}) + 4 p^{3} (- 2 n) + 4 \cdot 3 p \cdot p$

Step 1.6

Simplify each term.

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Step 1.6.1

Rewrite using the commutative property of multiplication.

$8 n^{4} p^{2} - 12 n^{2} p^{2} + 20 n^{2} + 4 \cdot 6 p^{2} n^{2} + 4 p^{3} (- 2 n) + 4 \cdot 3 p \cdot p$

Step 1.6.2

Multiply $4$ by $6$ .

$8 n^{4} p^{2} - 12 n^{2} p^{2} + 20 n^{2} + 24 p^{2} n^{2} + 4 p^{3} (- 2 n) + 4 \cdot 3 p \cdot p$

Step 1.6.3

Rewrite using the commutative property of multiplication.

$8 n^{4} p^{2} - 12 n^{2} p^{2} + 20 n^{2} + 24 p^{2} n^{2} + 4 \cdot - 2 p^{3} n + 4 \cdot 3 p \cdot p$

Step 1.6.4

Multiply $4$ by $- 2$ .

$8 n^{4} p^{2} - 12 n^{2} p^{2} + 20 n^{2} + 24 p^{2} n^{2} - 8 p^{3} n + 4 \cdot 3 p \cdot p$

Step 1.6.5

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

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Step 1.6.5.1

Move $p$ .

$8 n^{4} p^{2} - 12 n^{2} p^{2} + 20 n^{2} + 24 p^{2} n^{2} - 8 p^{3} n + 4 \cdot 3 (p \cdot p)$

Step 1.6.5.2

Multiply $p$ by $p$ .

$8 n^{4} p^{2} - 12 n^{2} p^{2} + 20 n^{2} + 24 p^{2} n^{2} - 8 p^{3} n + 4 \cdot 3 p^{2}$

Step 1.6.6

Multiply $4$ by $3$ .

$8 n^{4} p^{2} - 12 n^{2} p^{2} + 20 n^{2} + 24 p^{2} n^{2} - 8 p^{3} n + 12 p^{2}$

Step 2

Add $- 12 n^{2} p^{2}$ and $24 p^{2} n^{2}$ .

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Step 2.1

Move $p^{2}$ .

$8 n^{4} p^{2} + 20 n^{2} - 12 n^{2} p^{2} + 24 n^{2} p^{2} - 8 p^{3} n + 12 p^{2}$

Step 2.2

Add $- 12 n^{2} p^{2}$ and $24 n^{2} p^{2}$ .

$8 n^{4} p^{2} + 20 n^{2} + 12 n^{2} p^{2} - 8 p^{3} n + 12 p^{2}$

Step 3

Move $p^{3}$ .

$8 n^{4} p^{2} + 20 n^{2} + 12 n^{2} p^{2} - 8 n p^{3} + 12 p^{2}$

Step 4

Move $20 n^{2}$ .

$8 n^{4} p^{2} + 12 n^{2} p^{2} - 8 n p^{3} + 12 p^{2} + 20 n^{2}$

Step 5

Move $12 n^{2} p^{2}$ .

$8 n^{4} p^{2} - 8 n p^{3} + 12 n^{2} p^{2} + 12 p^{2} + 20 n^{2}$