Algebra Examples

$\sqrt{{({(a + b)}^{2} - {(a - b)}^{2})}^{2} + a^{4} - 12 a^{2} b^{2} + 4 b^{4}}$

Step 1

Simplify each term.

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Step 1.1

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

$\sqrt{{((a + b) (a + b) - {(a - b)}^{2})}^{2} + a^{4} - 12 a^{2} b^{2} + 4 b^{4}}$

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

$\sqrt{{(a (a + b) + b (a + b) - {(a - b)}^{2})}^{2} + a^{4} - 12 a^{2} b^{2} + 4 b^{4}}$

Step 1.2.2

Apply the distributive property.

$\sqrt{{(a \cdot a + a b + b (a + b) - {(a - b)}^{2})}^{2} + a^{4} - 12 a^{2} b^{2} + 4 b^{4}}$

Step 1.2.3

Apply the distributive property.

$\sqrt{{(a \cdot a + a b + b a + b \cdot b - {(a - b)}^{2})}^{2} + a^{4} - 12 a^{2} b^{2} + 4 b^{4}}$

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 $a$ .

$\sqrt{{(a^{2} + a b + b a + b \cdot b - {(a - b)}^{2})}^{2} + a^{4} - 12 a^{2} b^{2} + 4 b^{4}}$

Step 1.3.1.2

Multiply $b$ by $b$ .

$\sqrt{{(a^{2} + a b + b a + b^{2} - {(a - b)}^{2})}^{2} + a^{4} - 12 a^{2} b^{2} + 4 b^{4}}$

Step 1.3.2

Add $a b$ and $b a$ .

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Step 1.3.2.1

Reorder $b$ and $a$ .

$\sqrt{{(a^{2} + a b + a b + b^{2} - {(a - b)}^{2})}^{2} + a^{4} - 12 a^{2} b^{2} + 4 b^{4}}$

Step 1.3.2.2

Add $a b$ and $a b$ .

$\sqrt{{(a^{2} + 2 a b + b^{2} - {(a - b)}^{2})}^{2} + a^{4} - 12 a^{2} b^{2} + 4 b^{4}}$

Step 1.4

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

$\sqrt{{(a^{2} + 2 a b + b^{2} - ((a - b) (a - b)))}^{2} + a^{4} - 12 a^{2} b^{2} + 4 b^{4}}$

Step 1.5

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

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Step 1.5.1

Apply the distributive property.

$\sqrt{{(a^{2} + 2 a b + b^{2} - (a (a - b) - b (a - b)))}^{2} + a^{4} - 12 a^{2} b^{2} + 4 b^{4}}$

Step 1.5.2

Apply the distributive property.

$\sqrt{{(a^{2} + 2 a b + b^{2} - (a \cdot a + a (- b) - b (a - b)))}^{2} + a^{4} - 12 a^{2} b^{2} + 4 b^{4}}$

Step 1.5.3

Apply the distributive property.

$\sqrt{{(a^{2} + 2 a b + b^{2} - (a \cdot a + a (- b) - b a - b (- b)))}^{2} + a^{4} - 12 a^{2} b^{2} + 4 b^{4}}$

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 $a$ .

$\sqrt{{(a^{2} + 2 a b + b^{2} - (a^{2} + a (- b) - b a - b (- b)))}^{2} + a^{4} - 12 a^{2} b^{2} + 4 b^{4}}$

Step 1.6.1.2

Rewrite using the commutative property of multiplication.

$\sqrt{{(a^{2} + 2 a b + b^{2} - (a^{2} - a b - b a - b (- b)))}^{2} + a^{4} - 12 a^{2} b^{2} + 4 b^{4}}$

Step 1.6.1.3

Rewrite using the commutative property of multiplication.

$\sqrt{{(a^{2} + 2 a b + b^{2} - (a^{2} - a b - b a - 1 \cdot - 1 b \cdot b))}^{2} + a^{4} - 12 a^{2} b^{2} + 4 b^{4}}$

Step 1.6.1.4

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

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Step 1.6.1.4.1

Move $b$ .

$\sqrt{{(a^{2} + 2 a b + b^{2} - (a^{2} - a b - b a - 1 \cdot - 1 (b \cdot b)))}^{2} + a^{4} - 12 a^{2} b^{2} + 4 b^{4}}$

Step 1.6.1.4.2

Multiply $b$ by $b$ .

$\sqrt{{(a^{2} + 2 a b + b^{2} - (a^{2} - a b - b a - 1 \cdot - 1 b^{2}))}^{2} + a^{4} - 12 a^{2} b^{2} + 4 b^{4}}$

Step 1.6.1.5

Multiply $- 1$ by $- 1$ .

$\sqrt{{(a^{2} + 2 a b + b^{2} - (a^{2} - a b - b a + 1 b^{2}))}^{2} + a^{4} - 12 a^{2} b^{2} + 4 b^{4}}$

Step 1.6.1.6

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

$\sqrt{{(a^{2} + 2 a b + b^{2} - (a^{2} - a b - b a + b^{2}))}^{2} + a^{4} - 12 a^{2} b^{2} + 4 b^{4}}$

Step 1.6.2

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

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Step 1.6.2.1

Move $b$ .

$\sqrt{{(a^{2} + 2 a b + b^{2} - (a^{2} - a b - 1 a b + b^{2}))}^{2} + a^{4} - 12 a^{2} b^{2} + 4 b^{4}}$

Step 1.6.2.2

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

$\sqrt{{(a^{2} + 2 a b + b^{2} - (a^{2} - 2 a b + b^{2}))}^{2} + a^{4} - 12 a^{2} b^{2} + 4 b^{4}}$

Step 1.7

Apply the distributive property.

$\sqrt{{(a^{2} + 2 a b + b^{2} - a^{2} - (- 2 a b) - b^{2})}^{2} + a^{4} - 12 a^{2} b^{2} + 4 b^{4}}$

Step 1.8

Multiply $- 2$ by $- 1$ .

$\sqrt{{(a^{2} + 2 a b + b^{2} - a^{2} + 2 a b - b^{2})}^{2} + a^{4} - 12 a^{2} b^{2} + 4 b^{4}}$

Step 2

Simplify by adding terms.

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Step 2.1

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

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Step 2.1.1

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

$\sqrt{{(2 a b + b^{2} + 0 + 2 a b - b^{2})}^{2} + a^{4} - 12 a^{2} b^{2} + 4 b^{4}}$

Step 2.1.2

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

$\sqrt{{(2 a b + b^{2} + 2 a b - b^{2})}^{2} + a^{4} - 12 a^{2} b^{2} + 4 b^{4}}$

Step 2.1.3

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

$\sqrt{{(2 a b + 2 a b + 0)}^{2} + a^{4} - 12 a^{2} b^{2} + 4 b^{4}}$

Step 2.1.4

Add $2 a b + 2 a b$ and $0$ .

$\sqrt{{(2 a b + 2 a b)}^{2} + a^{4} - 12 a^{2} b^{2} + 4 b^{4}}$

Step 2.2

Add $2 a b$ and $2 a b$ .

$\sqrt{{(4 a b)}^{2} + a^{4} - 12 a^{2} b^{2} + 4 b^{4}}$

Step 3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.1

Apply the product rule to $4 a b$ .

$\sqrt{{(4 a)}^{2} b^{2} + a^{4} - 12 a^{2} b^{2} + 4 b^{4}}$

Step 3.2

Apply the product rule to $4 a$ .

$\sqrt{4^{2} a^{2} b^{2} + a^{4} - 12 a^{2} b^{2} + 4 b^{4}}$

Step 4

Simplify by adding terms.

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Step 4.1

Raise $4$ to the power of $2$ .

$\sqrt{16 a^{2} b^{2} + a^{4} - 12 a^{2} b^{2} + 4 b^{4}}$

Step 4.2

Subtract $12 a^{2} b^{2}$ from $16 a^{2} b^{2}$ .

$\sqrt{a^{4} + 4 a^{2} b^{2} + 4 b^{4}}$

Step 5

Factor using the perfect square rule.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

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

Step 5.4

Rewrite the polynomial.

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

Step 5.5

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = a^{2}$ and $b = 2 b^{2}$ .

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

Step 6

Pull terms out from under the radical, assuming positive real numbers.

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