Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

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

Multiply $a$ by $a$ .

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

Step 1.3.1.2

Rewrite using the commutative property of multiplication.

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

Step 1.3.1.3

Rewrite using the commutative property of multiplication.

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

Step 1.3.1.4

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

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Step 1.3.1.4.1

Move $b$ .

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

Step 1.3.1.4.2

Multiply $b$ by $b$ .

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

Step 1.3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 1.3.1.6

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

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

Step 1.3.2

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

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Step 1.3.2.1

Move $b$ .

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

Step 1.3.2.2

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

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

Step 1.4

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

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

Step 1.5

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

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Step 1.5.1

Apply the distributive property.

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

Step 1.5.2

Apply the distributive property.

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

Step 1.5.3

Apply the distributive property.

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

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

Multiply $a$ by $a$ .

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

Step 1.6.1.2

Multiply $b$ by $b$ .

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

Step 1.6.2

Add $a b$ and $b a$ .

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Step 1.6.2.1

Reorder $b$ and $a$ .

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

Step 1.6.2.2

Add $a b$ and $a b$ .

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

Step 1.7

Expand $(a^{2} - 2 a b + b^{2}) (a^{2} + 2 a b + b^{2})$ by multiplying each term in the first expression by each term in the second expression.

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

Step 1.8

Combine the opposite terms in $a^{2} a^{2} + a^{2} (2 a b) + a^{2} b^{2} - 2 a b a^{2} - 2 a b (2 a b) - 2 a b \cdot b^{2} + b^{2} a^{2} + b^{2} (2 a b) + b^{2} b^{2}$ .

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Step 1.8.1

Reorder the factors in the terms $a^{2} (2 a b)$ and $- 2 a b a^{2}$ .

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

Step 1.8.2

Subtract $2 a^{2} a b$ from $2 a^{2} a b$ .

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

Step 1.8.3

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

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

Step 1.8.4

Reorder the factors in the terms $- 2 a b \cdot b^{2}$ and $b^{2} (2 a b)$ .

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

Step 1.8.5

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

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

Step 1.8.6

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

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

Step 1.9

Simplify each term.

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Step 1.9.1

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

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Step 1.9.1.1

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

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

Step 1.9.1.2

Add $2$ and $2$ .

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

Step 1.9.2

Multiply $a$ by $a$ by adding the exponents.

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Step 1.9.2.1

Move $a$ .

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

Step 1.9.2.2

Multiply $a$ by $a$ .

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

Step 1.9.3

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

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Step 1.9.3.1

Move $b$ .

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

Step 1.9.3.2

Multiply $b$ by $b$ .

$a^{4} + a^{2} b^{2} - 2 a^{2} b^{2} \cdot 2 + b^{2} a^{2} + b^{2} b^{2} + 2 a^{2} b^{2}$

Step 1.9.4

Multiply $2$ by $- 2$ .

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

Step 1.9.5

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

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Step 1.9.5.1

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

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

Step 1.9.5.2

Add $2$ and $2$ .

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

Step 1.10

Subtract $4 a^{2} b^{2}$ from $a^{2} b^{2}$ .

$a^{4} - 3 a^{2} b^{2} + b^{2} a^{2} + b^{4} + 2 a^{2} b^{2}$

Step 1.11

Add $- 3 a^{2} b^{2}$ and $b^{2} a^{2}$ .

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Step 1.11.1

Reorder $b^{2}$ and $a^{2}$ .

$a^{4} - 3 a^{2} b^{2} + a^{2} b^{2} + b^{4} + 2 a^{2} b^{2}$

Step 1.11.2

Add $- 3 a^{2} b^{2}$ and $a^{2} b^{2}$ .

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

Step 2

Combine the opposite terms in $a^{4} - 2 a^{2} b^{2} + b^{4} + 2 a^{2} b^{2}$ .

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Step 2.1

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

$a^{4} + b^{4} + 0$

Step 2.2

Add $a^{4} + b^{4}$ and $0$ .

$a^{4} + b^{4}$