Basic Math Examples

$(16 m - {(a + b)}^{2 n}) (5 m + 3 {(a + b)}^{2})$

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(a + b)}^{2}$ as $(a + b) (a + b)$ .

$(16 m - {(a + b)}^{2 n}) (5 m + 3 ((a + b) (a + b)))$

Step 1.2

Expand $(a + b) (a + b)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$(16 m - {(a + b)}^{2 n}) (5 m + 3 (a (a + b) + b (a + b)))$

Step 1.2.2

Apply the distributive property.

$(16 m - {(a + b)}^{2 n}) (5 m + 3 (a \cdot a + a b + b (a + b)))$

Step 1.2.3

Apply the distributive property.

$(16 m - {(a + b)}^{2 n}) (5 m + 3 (a \cdot a + a b + b a + b \cdot b))$

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

Multiply $a$ by $a$ .

$(16 m - {(a + b)}^{2 n}) (5 m + 3 (a^{2} + a b + b a + b \cdot b))$

Step 1.3.1.2

Multiply $b$ by $b$ .

$(16 m - {(a + b)}^{2 n}) (5 m + 3 (a^{2} + a b + b a + b^{2}))$

Step 1.3.2

Add $a b$ and $b a$ .

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Step 1.3.2.1

Reorder $b$ and $a$ .

$(16 m - {(a + b)}^{2 n}) (5 m + 3 (a^{2} + a b + a b + b^{2}))$

Step 1.3.2.2

Add $a b$ and $a b$ .

$(16 m - {(a + b)}^{2 n}) (5 m + 3 (a^{2} + 2 a b + b^{2}))$

Step 1.4

Apply the distributive property.

$(16 m - {(a + b)}^{2 n}) (5 m + 3 a^{2} + 3 (2 a b) + 3 b^{2})$

Step 1.5

Multiply $2$ by $3$ .

$(16 m - {(a + b)}^{2 n}) (5 m + 3 a^{2} + 6 a b + 3 b^{2})$

Step 2

Expand $(16 m - {(a + b)}^{2 n}) (5 m + 3 a^{2} + 6 a b + 3 b^{2})$ by multiplying each term in the first expression by each term in the second expression.

$16 m (5 m) + 16 m (3 a^{2}) + 16 m (6 a b) + 16 m (3 b^{2}) - {(a + b)}^{2 n} (5 m) - {(a + b)}^{2 n} (3 a^{2}) - {(a + b)}^{2 n} (6 a b) - {(a + b)}^{2 n} (3 b^{2})$

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite using the commutative property of multiplication.

$16 \cdot 5 m \cdot m + 16 m (3 a^{2}) + 16 m (6 a b) + 16 m (3 b^{2}) - {(a + b)}^{2 n} (5 m) - {(a + b)}^{2 n} (3 a^{2}) - {(a + b)}^{2 n} (6 a b) - {(a + b)}^{2 n} (3 b^{2})$

Step 3.1.2

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

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Step 3.1.2.1

Move $m$ .

$16 \cdot 5 (m \cdot m) + 16 m (3 a^{2}) + 16 m (6 a b) + 16 m (3 b^{2}) - {(a + b)}^{2 n} (5 m) - {(a + b)}^{2 n} (3 a^{2}) - {(a + b)}^{2 n} (6 a b) - {(a + b)}^{2 n} (3 b^{2})$

Step 3.1.2.2

Multiply $m$ by $m$ .

$16 \cdot 5 m^{2} + 16 m (3 a^{2}) + 16 m (6 a b) + 16 m (3 b^{2}) - {(a + b)}^{2 n} (5 m) - {(a + b)}^{2 n} (3 a^{2}) - {(a + b)}^{2 n} (6 a b) - {(a + b)}^{2 n} (3 b^{2})$

Step 3.1.3

Multiply $16$ by $5$ .

$80 m^{2} + 16 m (3 a^{2}) + 16 m (6 a b) + 16 m (3 b^{2}) - {(a + b)}^{2 n} (5 m) - {(a + b)}^{2 n} (3 a^{2}) - {(a + b)}^{2 n} (6 a b) - {(a + b)}^{2 n} (3 b^{2})$

Step 3.1.4

Rewrite using the commutative property of multiplication.

$80 m^{2} + 16 \cdot 3 m a^{2} + 16 m (6 a b) + 16 m (3 b^{2}) - {(a + b)}^{2 n} (5 m) - {(a + b)}^{2 n} (3 a^{2}) - {(a + b)}^{2 n} (6 a b) - {(a + b)}^{2 n} (3 b^{2})$

Step 3.1.5

Multiply $16$ by $3$ .

$80 m^{2} + 48 m a^{2} + 16 m (6 a b) + 16 m (3 b^{2}) - {(a + b)}^{2 n} (5 m) - {(a + b)}^{2 n} (3 a^{2}) - {(a + b)}^{2 n} (6 a b) - {(a + b)}^{2 n} (3 b^{2})$

Step 3.1.6

Multiply $6$ by $16$ .

$80 m^{2} + 48 m a^{2} + 96 m (a b) + 16 m (3 b^{2}) - {(a + b)}^{2 n} (5 m) - {(a + b)}^{2 n} (3 a^{2}) - {(a + b)}^{2 n} (6 a b) - {(a + b)}^{2 n} (3 b^{2})$

Step 3.1.7

Rewrite using the commutative property of multiplication.

$80 m^{2} + 48 m a^{2} + 96 m a b + 16 \cdot 3 m b^{2} - {(a + b)}^{2 n} (5 m) - {(a + b)}^{2 n} (3 a^{2}) - {(a + b)}^{2 n} (6 a b) - {(a + b)}^{2 n} (3 b^{2})$

Step 3.1.8

Multiply $16$ by $3$ .

$80 m^{2} + 48 m a^{2} + 96 m a b + 48 m b^{2} - {(a + b)}^{2 n} (5 m) - {(a + b)}^{2 n} (3 a^{2}) - {(a + b)}^{2 n} (6 a b) - {(a + b)}^{2 n} (3 b^{2})$

Step 3.1.9

Rewrite using the commutative property of multiplication.

$80 m^{2} + 48 m a^{2} + 96 m a b + 48 m b^{2} - 1 \cdot 5 {(a + b)}^{2 n} m - {(a + b)}^{2 n} (3 a^{2}) - {(a + b)}^{2 n} (6 a b) - {(a + b)}^{2 n} (3 b^{2})$

Step 3.1.10

Multiply $- 1$ by $5$ .

$80 m^{2} + 48 m a^{2} + 96 m a b + 48 m b^{2} - 5 {(a + b)}^{2 n} m - {(a + b)}^{2 n} (3 a^{2}) - {(a + b)}^{2 n} (6 a b) - {(a + b)}^{2 n} (3 b^{2})$

Step 3.1.11

Rewrite using the commutative property of multiplication.

$80 m^{2} + 48 m a^{2} + 96 m a b + 48 m b^{2} - 5 {(a + b)}^{2 n} m - 1 \cdot 3 {(a + b)}^{2 n} a^{2} - {(a + b)}^{2 n} (6 a b) - {(a + b)}^{2 n} (3 b^{2})$

Step 3.1.12

Multiply $- 1$ by $3$ .

$80 m^{2} + 48 m a^{2} + 96 m a b + 48 m b^{2} - 5 {(a + b)}^{2 n} m - 3 {(a + b)}^{2 n} a^{2} - {(a + b)}^{2 n} (6 a b) - {(a + b)}^{2 n} (3 b^{2})$

Step 3.1.13

Multiply $6$ by $- 1$ .

$80 m^{2} + 48 m a^{2} + 96 m a b + 48 m b^{2} - 5 {(a + b)}^{2 n} m - 3 {(a + b)}^{2 n} a^{2} - 6 {(a + b)}^{2 n} (a b) - {(a + b)}^{2 n} (3 b^{2})$

Step 3.1.14

Rewrite using the commutative property of multiplication.

$80 m^{2} + 48 m a^{2} + 96 m a b + 48 m b^{2} - 5 {(a + b)}^{2 n} m - 3 {(a + b)}^{2 n} a^{2} - 6 {(a + b)}^{2 n} (a b) - 1 \cdot 3 {(a + b)}^{2 n} b^{2}$

Step 3.1.15

Multiply $- 1$ by $3$ .

$80 m^{2} + 48 m a^{2} + 96 m a b + 48 m b^{2} - 5 {(a + b)}^{2 n} m - 3 {(a + b)}^{2 n} a^{2} - 6 {(a + b)}^{2 n} (a b) - 3 {(a + b)}^{2 n} b^{2}$

Step 3.2

Reorder factors in $80 m^{2} + 48 m a^{2} + 96 m a b + 48 m b^{2} - 5 {(a + b)}^{2 n} m - 3 {(a + b)}^{2 n} a^{2} - 6 {(a + b)}^{2 n} (a b) - 3 {(a + b)}^{2 n} b^{2}$ .

$80 m^{2} + 48 m a^{2} + 96 m a b + 48 m b^{2} - 5 m {(a + b)}^{2 n} - 3 a^{2} {(a + b)}^{2 n} - 6 a b {(a + b)}^{2 n} - 3 b^{2} {(a + b)}^{2 n}$