Algebra Examples

$(3 m - 2 n^{2}) (2 m^{2} + 3 n)$

Step 1

Expand $(3 m - 2 n^{2}) (2 m^{2} + 3 n)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$3 m (2 m^{2} + 3 n) - 2 n^{2} (2 m^{2} + 3 n)$

Step 1.2

Apply the distributive property.

$3 m (2 m^{2}) + 3 m (3 n) - 2 n^{2} (2 m^{2} + 3 n)$

Step 1.3

Apply the distributive property.

$3 m (2 m^{2}) + 3 m (3 n) - 2 n^{2} (2 m^{2}) - 2 n^{2} (3 n)$

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.

$3 \cdot 2 m \cdot m^{2} + 3 m (3 n) - 2 n^{2} (2 m^{2}) - 2 n^{2} (3 n)$

Step 2.1.2

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

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Step 2.1.2.1

Move $m^{2}$ .

$3 \cdot 2 (m^{2} m) + 3 m (3 n) - 2 n^{2} (2 m^{2}) - 2 n^{2} (3 n)$

Step 2.1.2.2

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

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Step 2.1.2.2.1

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

$3 \cdot 2 (m^{2} m^{1}) + 3 m (3 n) - 2 n^{2} (2 m^{2}) - 2 n^{2} (3 n)$

Step 2.1.2.2.2

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

$3 \cdot 2 m^{2 + 1} + 3 m (3 n) - 2 n^{2} (2 m^{2}) - 2 n^{2} (3 n)$

Step 2.1.2.3

Add $2$ and $1$ .

$3 \cdot 2 m^{3} + 3 m (3 n) - 2 n^{2} (2 m^{2}) - 2 n^{2} (3 n)$

Step 2.1.3

Multiply $3$ by $2$ .

$6 m^{3} + 3 m (3 n) - 2 n^{2} (2 m^{2}) - 2 n^{2} (3 n)$

Step 2.1.4

Rewrite using the commutative property of multiplication.

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

Step 2.1.5

Multiply $3$ by $3$ .

$6 m^{3} + 9 m n - 2 n^{2} (2 m^{2}) - 2 n^{2} (3 n)$

Step 2.1.6

Rewrite using the commutative property of multiplication.

$6 m^{3} + 9 m n - 2 \cdot 2 n^{2} m^{2} - 2 n^{2} (3 n)$

Step 2.1.7

Multiply $- 2$ by $2$ .

$6 m^{3} + 9 m n - 4 n^{2} m^{2} - 2 n^{2} (3 n)$

Step 2.1.8

Rewrite using the commutative property of multiplication.

$6 m^{3} + 9 m n - 4 n^{2} m^{2} - 2 \cdot 3 n^{2} n$

Step 2.1.9

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

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Step 2.1.9.1

Move $n$ .

$6 m^{3} + 9 m n - 4 n^{2} m^{2} - 2 \cdot 3 (n \cdot n^{2})$

Step 2.1.9.2

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

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Step 2.1.9.2.1

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

$6 m^{3} + 9 m n - 4 n^{2} m^{2} - 2 \cdot 3 (n^{1} n^{2})$

Step 2.1.9.2.2

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

$6 m^{3} + 9 m n - 4 n^{2} m^{2} - 2 \cdot 3 n^{1 + 2}$

Step 2.1.9.3

Add $1$ and $2$ .

$6 m^{3} + 9 m n - 4 n^{2} m^{2} - 2 \cdot 3 n^{3}$

Step 2.1.10

Multiply $- 2$ by $3$ .

$6 m^{3} + 9 m n - 4 n^{2} m^{2} - 6 n^{3}$

Step 2.2

Simplify the expression.

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Step 2.2.1

Move $n^{2}$ .

$6 m^{3} + 9 m n - 4 m^{2} n^{2} - 6 n^{3}$

Step 2.2.2

Move $9 m n$ .

$6 m^{3} - 4 m^{2} n^{2} - 6 n^{3} + 9 m n$

Step 2.2.3

Reorder $6 m^{3}$ and $- 4 m^{2} n^{2}$ .

$- 4 m^{2} n^{2} + 6 m^{3} - 6 n^{3} + 9 m n$