Algebra Examples

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

Step 1

Apply the distributive property.

$- 4 m n^{3} (- 3 m^{2} n) - 4 m n^{3} (12 m n^{2}) - 4 m n^{3} (- 6 m) - 4 m n^{3} (7 n^{2})$

Step 2

Simplify.

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Step 2.1

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

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Step 2.1.1

Move $m^{2}$ .

$- 4 (m^{2} m) n^{3} (- 3 n) - 4 m n^{3} (12 m n^{2}) - 4 m n^{3} (- 6 m) - 4 m n^{3} (7 n^{2})$

Step 2.1.2

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

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Step 2.1.2.1

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

$- 4 (m^{2} m^{1}) n^{3} (- 3 n) - 4 m n^{3} (12 m n^{2}) - 4 m n^{3} (- 6 m) - 4 m n^{3} (7 n^{2})$

Step 2.1.2.2

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

$- 4 m^{2 + 1} n^{3} (- 3 n) - 4 m n^{3} (12 m n^{2}) - 4 m n^{3} (- 6 m) - 4 m n^{3} (7 n^{2})$

Step 2.1.3

Add $2$ and $1$ .

$- 4 m^{3} n^{3} (- 3 n) - 4 m n^{3} (12 m n^{2}) - 4 m n^{3} (- 6 m) - 4 m n^{3} (7 n^{2})$

Step 2.2

Multiply $m$ by $m$ by adding the exponents.

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Step 2.2.1

Move $m$ .

$- 4 m^{3} n^{3} (- 3 n) - 4 (m \cdot m) n^{3} (12 n^{2}) - 4 m n^{3} (- 6 m) - 4 m n^{3} (7 n^{2})$

Step 2.2.2

Multiply $m$ by $m$ .

$- 4 m^{3} n^{3} (- 3 n) - 4 m^{2} n^{3} (12 n^{2}) - 4 m n^{3} (- 6 m) - 4 m n^{3} (7 n^{2})$

Step 2.3

Multiply $m$ by $m$ by adding the exponents.

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Step 2.3.1

Move $m$ .

$- 4 m^{3} n^{3} (- 3 n) - 4 m^{2} n^{3} (12 n^{2}) - 4 (m \cdot m) n^{3} \cdot - 6 - 4 m n^{3} (7 n^{2})$

Step 2.3.2

Multiply $m$ by $m$ .

$- 4 m^{3} n^{3} (- 3 n) - 4 m^{2} n^{3} (12 n^{2}) - 4 m^{2} n^{3} \cdot - 6 - 4 m n^{3} (7 n^{2})$

Step 2.4

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

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Step 2.4.1

Move $n^{2}$ .

$- 4 m^{3} n^{3} (- 3 n) - 4 m^{2} n^{3} (12 n^{2}) - 4 m^{2} n^{3} \cdot - 6 - 4 m (n^{2} n^{3}) \cdot 7$

Step 2.4.2

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

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

Step 2.4.3

Add $2$ and $3$ .

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

Step 3

Simplify each term.

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Step 3.1

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

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Step 3.1.1

Move $n$ .

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

Step 3.1.2

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

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Step 3.1.2.1

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

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

Step 3.1.2.2

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

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

Step 3.1.3

Add $1$ and $3$ .

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

Step 3.2

Multiply $- 3$ by $- 4$ .

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

Step 3.3

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

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Step 3.3.1

Move $n^{2}$ .

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

Step 3.3.2

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

$12 m^{3} n^{4} - 4 m^{2} n^{2 + 3} \cdot 12 - 4 m^{2} n^{3} \cdot - 6 - 4 m n^{5} \cdot 7$

Step 3.3.3

Add $2$ and $3$ .

$12 m^{3} n^{4} - 4 m^{2} n^{5} \cdot 12 - 4 m^{2} n^{3} \cdot - 6 - 4 m n^{5} \cdot 7$

Step 3.4

Multiply $12$ by $- 4$ .

$12 m^{3} n^{4} - 48 m^{2} n^{5} - 4 m^{2} n^{3} \cdot - 6 - 4 m n^{5} \cdot 7$

Step 3.5

Multiply $- 6$ by $- 4$ .

$12 m^{3} n^{4} - 48 m^{2} n^{5} + 24 m^{2} n^{3} - 4 m n^{5} \cdot 7$

Step 3.6

Multiply $7$ by $- 4$ .

$12 m^{3} n^{4} - 48 m^{2} n^{5} + 24 m^{2} n^{3} - 28 m n^{5}$