Algebra Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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Step 2.1.1.1

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

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

Step 2.1.1.2

Add $2$ and $2$ .

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

Step 2.1.2

Rewrite using the commutative property of multiplication.

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

Step 2.1.3

Rewrite using the commutative property of multiplication.

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

Step 2.1.4

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

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Step 2.1.4.1

Move $n^{2}$ .

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

Step 2.1.4.2

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

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Step 2.1.4.2.1

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

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

Step 2.1.4.2.2

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

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

Step 2.1.4.3

Add $2$ and $1$ .

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

Step 2.1.5

Multiply $- 1$ by $2$ .

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

Step 2.2

Simplify the expression.

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Step 2.2.1

Move $n$ .

$m^{4} + 2 m^{2} n^{2} - 1 m^{2} n - 2 n^{3}$

Step 2.2.2

Move $- 1 m^{2} n$ .

$m^{4} + 2 m^{2} n^{2} - 2 n^{3} - 1 m^{2} n$

Step 2.2.3

Reorder $m^{4}$ and $2 m^{2} n^{2}$ .

$2 m^{2} n^{2} + m^{4} - 2 n^{3} - 1 m^{2} n$