Algebra Examples

${(a + b)}^{4} - {(a - b)}^{4}$

Step 1

Rewrite ${(a + b)}^{4}$ as ${({(a + b)}^{2})}^{2}$ .

${({(a + b)}^{2})}^{2} - {(a - b)}^{4}$

Step 2

Rewrite ${(a - b)}^{4}$ as ${({(a - b)}^{2})}^{2}$ .

${({(a + b)}^{2})}^{2} - {({(a - b)}^{2})}^{2}$

Step 3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = {(a + b)}^{2}$ and $b = {(a - b)}^{2}$ .

$({(a + b)}^{2} + {(a - b)}^{2}) ({(a + b)}^{2} - {(a - b)}^{2})$

Step 4

Simplify.

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Step 4.1

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

$((a + b) (a + b) + {(a - b)}^{2}) ({(a + b)}^{2} - {(a - b)}^{2})$

Step 4.2

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

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Step 4.2.1

Apply the distributive property.

$(a (a + b) + b (a + b) + {(a - b)}^{2}) ({(a + b)}^{2} - {(a - b)}^{2})$

Step 4.2.2

Apply the distributive property.

$(a \cdot a + a b + b (a + b) + {(a - b)}^{2}) ({(a + b)}^{2} - {(a - b)}^{2})$

Step 4.2.3

Apply the distributive property.

$(a \cdot a + a b + b a + b \cdot b + {(a - b)}^{2}) ({(a + b)}^{2} - {(a - b)}^{2})$

Step 4.3

Simplify and combine like terms.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply $a$ by $a$ .

$(a^{2} + a b + b a + b \cdot b + {(a - b)}^{2}) ({(a + b)}^{2} - {(a - b)}^{2})$

Step 4.3.1.2

Multiply $b$ by $b$ .

$(a^{2} + a b + b a + b^{2} + {(a - b)}^{2}) ({(a + b)}^{2} - {(a - b)}^{2})$

Step 4.3.2

Add $a b$ and $b a$ .

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Step 4.3.2.1

Reorder $b$ and $a$ .

$(a^{2} + a b + a b + b^{2} + {(a - b)}^{2}) ({(a + b)}^{2} - {(a - b)}^{2})$

Step 4.3.2.2

Add $a b$ and $a b$ .

$(a^{2} + 2 a b + b^{2} + {(a - b)}^{2}) ({(a + b)}^{2} - {(a - b)}^{2})$

Step 4.4

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

$(a^{2} + 2 a b + b^{2} + (a - b) (a - b)) ({(a + b)}^{2} - {(a - b)}^{2})$

Step 4.5

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

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Step 4.5.1

Apply the distributive property.

$(a^{2} + 2 a b + b^{2} + a (a - b) - b (a - b)) ({(a + b)}^{2} - {(a - b)}^{2})$

Step 4.5.2

Apply the distributive property.

$(a^{2} + 2 a b + b^{2} + a \cdot a + a (- b) - b (a - b)) ({(a + b)}^{2} - {(a - b)}^{2})$

Step 4.5.3

Apply the distributive property.

$(a^{2} + 2 a b + b^{2} + a \cdot a + a (- b) - b a - b (- b)) ({(a + b)}^{2} - {(a - b)}^{2})$

Step 4.6

Simplify and combine like terms.

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Step 4.6.1

Simplify each term.

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Step 4.6.1.1

Multiply $a$ by $a$ .

$(a^{2} + 2 a b + b^{2} + a^{2} + a (- b) - b a - b (- b)) ({(a + b)}^{2} - {(a - b)}^{2})$

Step 4.6.1.2

Rewrite using the commutative property of multiplication.

$(a^{2} + 2 a b + b^{2} + a^{2} - a b - b a - b (- b)) ({(a + b)}^{2} - {(a - b)}^{2})$

Step 4.6.1.3

Rewrite using the commutative property of multiplication.

$(a^{2} + 2 a b + b^{2} + a^{2} - a b - b a - 1 \cdot - 1 b \cdot b) ({(a + b)}^{2} - {(a - b)}^{2})$

Step 4.6.1.4

Multiply $b$ by $b$ by adding the exponents.

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Step 4.6.1.4.1

Move $b$ .

$(a^{2} + 2 a b + b^{2} + a^{2} - a b - b a - 1 \cdot - 1 (b \cdot b)) ({(a + b)}^{2} - {(a - b)}^{2})$

Step 4.6.1.4.2

Multiply $b$ by $b$ .

$(a^{2} + 2 a b + b^{2} + a^{2} - a b - b a - 1 \cdot - 1 b^{2}) ({(a + b)}^{2} - {(a - b)}^{2})$

Step 4.6.1.5

Multiply $- 1$ by $- 1$ .

$(a^{2} + 2 a b + b^{2} + a^{2} - a b - b a + 1 b^{2}) ({(a + b)}^{2} - {(a - b)}^{2})$

Step 4.6.1.6

Multiply $b^{2}$ by $1$ .

$(a^{2} + 2 a b + b^{2} + a^{2} - a b - b a + b^{2}) ({(a + b)}^{2} - {(a - b)}^{2})$

Step 4.6.2

Subtract $b a$ from $- a b$ .

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Step 4.6.2.1

Move $b$ .

$(a^{2} + 2 a b + b^{2} + a^{2} - a b - 1 a b + b^{2}) ({(a + b)}^{2} - {(a - b)}^{2})$

Step 4.6.2.2

Subtract $a b$ from $- a b$ .

$(a^{2} + 2 a b + b^{2} + a^{2} - 2 a b + b^{2}) ({(a + b)}^{2} - {(a - b)}^{2})$

Step 4.7

Add $a^{2}$ and $a^{2}$ .

$(2 a^{2} + 2 a b + b^{2} - 2 a b + b^{2}) ({(a + b)}^{2} - {(a - b)}^{2})$

Step 4.8

Subtract $2 a b$ from $2 a b$ .

$(2 a^{2} + b^{2} + 0 + b^{2}) ({(a + b)}^{2} - {(a - b)}^{2})$

Step 4.9

Add $2 a^{2}$ and $0$ .

$(2 a^{2} + b^{2} + b^{2}) ({(a + b)}^{2} - {(a - b)}^{2})$

Step 4.10

Add $b^{2}$ and $b^{2}$ .

$(2 a^{2} + 2 b^{2}) ({(a + b)}^{2} - {(a - b)}^{2})$

Step 4.11

Factor $2$ out of $2 a^{2} + 2 b^{2}$ .

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Step 4.11.1

Factor $2$ out of $2 a^{2}$ .

$(2 (a^{2}) + 2 b^{2}) ({(a + b)}^{2} - {(a - b)}^{2})$

Step 4.11.2

Factor $2$ out of $2 b^{2}$ .

$(2 (a^{2}) + 2 (b^{2})) ({(a + b)}^{2} - {(a - b)}^{2})$

Step 4.11.3

Factor $2$ out of $2 (a^{2}) + 2 (b^{2})$ .

$2 (a^{2} + b^{2}) ({(a + b)}^{2} - {(a - b)}^{2})$

Step 4.12

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = a + b$ and $b = a - b$ .

$2 (a^{2} + b^{2}) ((a + b + a - b) (a + b - (a - b)))$

Step 4.13

Factor.

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Step 4.13.1

Simplify.

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Step 4.13.1.1

Remove unnecessary parentheses.

$2 (a^{2} + b^{2}) ((a + b + a - b) (a + b - (a - b)))$

Step 4.13.1.2

Add $a$ and $a$ .

$2 (a^{2} + b^{2}) ((2 a + b - b) (a + b - (a - b)))$

Step 4.13.1.3

Subtract $b$ from $b$ .

$2 (a^{2} + b^{2}) ((2 a + 0) (a + b - (a - b)))$

Step 4.13.1.4

Add $2 a$ and $0$ .

$2 (a^{2} + b^{2}) (2 a (a + b - (a - b)))$

Step 4.13.1.5

Apply the distributive property.

$2 (a^{2} + b^{2}) (2 a (a + b - a - - b))$

Step 4.13.1.6

Multiply $- - b$ .

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Step 4.13.1.6.1

Multiply $- 1$ by $- 1$ .

$2 (a^{2} + b^{2}) (2 a (a + b - a + 1 b))$

Step 4.13.1.6.2

Multiply $b$ by $1$ .

$2 (a^{2} + b^{2}) (2 a (a + b - a + b))$

Step 4.13.1.7

Subtract $a$ from $a$ .

$2 (a^{2} + b^{2}) (2 a (b + 0 + b))$

Step 4.13.1.8

Add $b$ and $0$ .

$2 (a^{2} + b^{2}) (2 a (b + b))$

Step 4.13.1.9

Factor.

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Step 4.13.1.9.1

Add $b$ and $b$ .

$2 (a^{2} + b^{2}) (2 a (2 b))$

Step 4.13.1.9.2

Remove unnecessary parentheses.

$2 (a^{2} + b^{2}) (2 a \cdot 2 b)$

Step 4.13.1.10

Multiply $2$ by $2$ .

$2 (a^{2} + b^{2}) (4 a b)$

Step 4.13.2

Remove unnecessary parentheses.

$2 (a^{2} + b^{2}) \cdot 4 a b$

Step 4.14

Multiply $4$ by $2$ .

$8 (a^{2} + b^{2}) a b$