Basic Math Examples

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

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(4 m - 2 n)}^{2}$ as $(4 m - 2 n) (4 m - 2 n)$ .

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

Step 1.2

Expand $(4 m - 2 n) (4 m - 2 n)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 1.3.1.2

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

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Step 1.3.1.2.1

Move $m$ .

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

Step 1.3.1.2.2

Multiply $m$ by $m$ .

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

Step 1.3.1.3

Multiply $4$ by $4$ .

$16 m^{2} + 4 m (- 2 n) - 2 n (4 m) - 2 n (- 2 n) - {(4 m + 2 n)}^{2}$

Step 1.3.1.4

Rewrite using the commutative property of multiplication.

$16 m^{2} + 4 \cdot - 2 m n - 2 n (4 m) - 2 n (- 2 n) - {(4 m + 2 n)}^{2}$

Step 1.3.1.5

Multiply $4$ by $- 2$ .

$16 m^{2} - 8 m n - 2 n (4 m) - 2 n (- 2 n) - {(4 m + 2 n)}^{2}$

Step 1.3.1.6

Rewrite using the commutative property of multiplication.

$16 m^{2} - 8 m n - 2 \cdot 4 n m - 2 n (- 2 n) - {(4 m + 2 n)}^{2}$

Step 1.3.1.7

Multiply $- 2$ by $4$ .

$16 m^{2} - 8 m n - 8 n m - 2 n (- 2 n) - {(4 m + 2 n)}^{2}$

Step 1.3.1.8

Rewrite using the commutative property of multiplication.

$16 m^{2} - 8 m n - 8 n m - 2 \cdot - 2 n \cdot n - {(4 m + 2 n)}^{2}$

Step 1.3.1.9

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

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Step 1.3.1.9.1

Move $n$ .

$16 m^{2} - 8 m n - 8 n m - 2 \cdot - 2 (n \cdot n) - {(4 m + 2 n)}^{2}$

Step 1.3.1.9.2

Multiply $n$ by $n$ .

$16 m^{2} - 8 m n - 8 n m - 2 \cdot - 2 n^{2} - {(4 m + 2 n)}^{2}$

Step 1.3.1.10

Multiply $- 2$ by $- 2$ .

$16 m^{2} - 8 m n - 8 n m + 4 n^{2} - {(4 m + 2 n)}^{2}$

Step 1.3.2

Subtract $8 n m$ from $- 8 m n$ .

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Step 1.3.2.1

Move $n$ .

$16 m^{2} - 8 m n - 8 m n + 4 n^{2} - {(4 m + 2 n)}^{2}$

Step 1.3.2.2

Subtract $8 m n$ from $- 8 m n$ .

$16 m^{2} - 16 m n + 4 n^{2} - {(4 m + 2 n)}^{2}$

Step 1.4

Rewrite ${(4 m + 2 n)}^{2}$ as $(4 m + 2 n) (4 m + 2 n)$ .

$16 m^{2} - 16 m n + 4 n^{2} - ((4 m + 2 n) (4 m + 2 n))$

Step 1.5

Expand $(4 m + 2 n) (4 m + 2 n)$ using the FOIL Method.

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Step 1.5.1

Apply the distributive property.

$16 m^{2} - 16 m n + 4 n^{2} - (4 m (4 m + 2 n) + 2 n (4 m + 2 n))$

Step 1.5.2

Apply the distributive property.

$16 m^{2} - 16 m n + 4 n^{2} - (4 m (4 m) + 4 m (2 n) + 2 n (4 m + 2 n))$

Step 1.5.3

Apply the distributive property.

$16 m^{2} - 16 m n + 4 n^{2} - (4 m (4 m) + 4 m (2 n) + 2 n (4 m) + 2 n (2 n))$

Step 1.6

Simplify and combine like terms.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

Rewrite using the commutative property of multiplication.

$16 m^{2} - 16 m n + 4 n^{2} - (4 \cdot 4 m \cdot m + 4 m (2 n) + 2 n (4 m) + 2 n (2 n))$

Step 1.6.1.2

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

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Step 1.6.1.2.1

Move $m$ .

$16 m^{2} - 16 m n + 4 n^{2} - (4 \cdot 4 (m \cdot m) + 4 m (2 n) + 2 n (4 m) + 2 n (2 n))$

Step 1.6.1.2.2

Multiply $m$ by $m$ .

$16 m^{2} - 16 m n + 4 n^{2} - (4 \cdot 4 m^{2} + 4 m (2 n) + 2 n (4 m) + 2 n (2 n))$

Step 1.6.1.3

Multiply $4$ by $4$ .

$16 m^{2} - 16 m n + 4 n^{2} - (16 m^{2} + 4 m (2 n) + 2 n (4 m) + 2 n (2 n))$

Step 1.6.1.4

Rewrite using the commutative property of multiplication.

$16 m^{2} - 16 m n + 4 n^{2} - (16 m^{2} + 4 \cdot 2 m n + 2 n (4 m) + 2 n (2 n))$

Step 1.6.1.5

Multiply $4$ by $2$ .

$16 m^{2} - 16 m n + 4 n^{2} - (16 m^{2} + 8 m n + 2 n (4 m) + 2 n (2 n))$

Step 1.6.1.6

Rewrite using the commutative property of multiplication.

$16 m^{2} - 16 m n + 4 n^{2} - (16 m^{2} + 8 m n + 2 \cdot 4 n m + 2 n (2 n))$

Step 1.6.1.7

Multiply $2$ by $4$ .

$16 m^{2} - 16 m n + 4 n^{2} - (16 m^{2} + 8 m n + 8 n m + 2 n (2 n))$

Step 1.6.1.8

Rewrite using the commutative property of multiplication.

$16 m^{2} - 16 m n + 4 n^{2} - (16 m^{2} + 8 m n + 8 n m + 2 \cdot 2 n \cdot n)$

Step 1.6.1.9

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

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Step 1.6.1.9.1

Move $n$ .

$16 m^{2} - 16 m n + 4 n^{2} - (16 m^{2} + 8 m n + 8 n m + 2 \cdot 2 (n \cdot n))$

Step 1.6.1.9.2

Multiply $n$ by $n$ .

$16 m^{2} - 16 m n + 4 n^{2} - (16 m^{2} + 8 m n + 8 n m + 2 \cdot 2 n^{2})$

Step 1.6.1.10

Multiply $2$ by $2$ .

$16 m^{2} - 16 m n + 4 n^{2} - (16 m^{2} + 8 m n + 8 n m + 4 n^{2})$

Step 1.6.2

Add $8 m n$ and $8 n m$ .

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Step 1.6.2.1

Move $n$ .

$16 m^{2} - 16 m n + 4 n^{2} - (16 m^{2} + 8 m n + 8 m n + 4 n^{2})$

Step 1.6.2.2

Add $8 m n$ and $8 m n$ .

$16 m^{2} - 16 m n + 4 n^{2} - (16 m^{2} + 16 m n + 4 n^{2})$

Step 1.7

Apply the distributive property.

$16 m^{2} - 16 m n + 4 n^{2} - (16 m^{2}) - (16 m n) - (4 n^{2})$

Step 1.8

Simplify.

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Step 1.8.1

Multiply $16$ by $- 1$ .

$16 m^{2} - 16 m n + 4 n^{2} - 16 m^{2} - (16 m n) - (4 n^{2})$

Step 1.8.2

Multiply $16$ by $- 1$ .

$16 m^{2} - 16 m n + 4 n^{2} - 16 m^{2} - 16 (m n) - (4 n^{2})$

Step 1.8.3

Multiply $4$ by $- 1$ .

$16 m^{2} - 16 m n + 4 n^{2} - 16 m^{2} - 16 (m n) - 4 n^{2}$

$16 m^{2} - 16 m n + 4 n^{2} - 16 m^{2} - 16 m n - 4 n^{2}$

Step 2

Simplify by adding terms.

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Step 2.1

Combine the opposite terms in $16 m^{2} - 16 m n + 4 n^{2} - 16 m^{2} - 16 m n - 4 n^{2}$ .

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Step 2.1.1

Subtract $16 m^{2}$ from $16 m^{2}$ .

$- 16 m n + 4 n^{2} + 0 - 16 m n - 4 n^{2}$

Step 2.1.2

Add $- 16 m n + 4 n^{2}$ and $0$ .

$- 16 m n + 4 n^{2} - 16 m n - 4 n^{2}$

Step 2.1.3

Subtract $4 n^{2}$ from $4 n^{2}$ .

$- 16 m n - 16 m n + 0$

Step 2.1.4

Add $- 16 m n - 16 m n$ and $0$ .

$- 16 m n - 16 m n$

Step 2.2

Subtract $16 m n$ from $- 16 m n$ .

$- 32 m n$