Pre-Algebra Examples

$\sqrt{16 a^{4} + 8 a^{2} b^{2} + b^{4} - 12 (2 a + b) (2 a - b)}$

Step 1

Apply the distributive property.

$\sqrt{16 a^{4} + 8 a^{2} b^{2} + b^{4} + (- 12 (2 a) - 12 b) (2 a - b)}$

Step 2

Multiply $2$ by $- 12$ .

$\sqrt{16 a^{4} + 8 a^{2} b^{2} + b^{4} + (- 24 a - 12 b) (2 a - b)}$

Step 3

Expand $(- 24 a - 12 b) (2 a - b)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\sqrt{16 a^{4} + 8 a^{2} b^{2} + b^{4} - 24 a (2 a - b) - 12 b (2 a - b)}$

Step 3.2

Apply the distributive property.

$\sqrt{16 a^{4} + 8 a^{2} b^{2} + b^{4} - 24 a (2 a) - 24 a (- b) - 12 b (2 a - b)}$

Step 3.3

Apply the distributive property.

$\sqrt{16 a^{4} + 8 a^{2} b^{2} + b^{4} - 24 a (2 a) - 24 a (- b) - 12 b (2 a) - 12 b (- b)}$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Rewrite using the commutative property of multiplication.

$\sqrt{16 a^{4} + 8 a^{2} b^{2} + b^{4} - 24 \cdot 2 a \cdot a - 24 a (- b) - 12 b (2 a) - 12 b (- b)}$

Step 4.1.2

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

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Step 4.1.2.1

Move $a$ .

$\sqrt{16 a^{4} + 8 a^{2} b^{2} + b^{4} - 24 \cdot 2 (a \cdot a) - 24 a (- b) - 12 b (2 a) - 12 b (- b)}$

Step 4.1.2.2

Multiply $a$ by $a$ .

$\sqrt{16 a^{4} + 8 a^{2} b^{2} + b^{4} - 24 \cdot 2 a^{2} - 24 a (- b) - 12 b (2 a) - 12 b (- b)}$

Step 4.1.3

Multiply $- 24$ by $2$ .

$\sqrt{16 a^{4} + 8 a^{2} b^{2} + b^{4} - 48 a^{2} - 24 a (- b) - 12 b (2 a) - 12 b (- b)}$

Step 4.1.4

Rewrite using the commutative property of multiplication.

$\sqrt{16 a^{4} + 8 a^{2} b^{2} + b^{4} - 48 a^{2} - 24 \cdot - 1 a b - 12 b (2 a) - 12 b (- b)}$

Step 4.1.5

Multiply $- 24$ by $- 1$ .

$\sqrt{16 a^{4} + 8 a^{2} b^{2} + b^{4} - 48 a^{2} + 24 a b - 12 b (2 a) - 12 b (- b)}$

Step 4.1.6

Rewrite using the commutative property of multiplication.

$\sqrt{16 a^{4} + 8 a^{2} b^{2} + b^{4} - 48 a^{2} + 24 a b - 12 \cdot 2 b a - 12 b (- b)}$

Step 4.1.7

Multiply $- 12$ by $2$ .

$\sqrt{16 a^{4} + 8 a^{2} b^{2} + b^{4} - 48 a^{2} + 24 a b - 24 b a - 12 b (- b)}$

Step 4.1.8

Rewrite using the commutative property of multiplication.

$\sqrt{16 a^{4} + 8 a^{2} b^{2} + b^{4} - 48 a^{2} + 24 a b - 24 b a - 12 \cdot - 1 b \cdot b}$

Step 4.1.9

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

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Step 4.1.9.1

Move $b$ .

$\sqrt{16 a^{4} + 8 a^{2} b^{2} + b^{4} - 48 a^{2} + 24 a b - 24 b a - 12 \cdot - 1 (b \cdot b)}$

Step 4.1.9.2

Multiply $b$ by $b$ .

$\sqrt{16 a^{4} + 8 a^{2} b^{2} + b^{4} - 48 a^{2} + 24 a b - 24 b a - 12 \cdot - 1 b^{2}}$

Step 4.1.10

Multiply $- 12$ by $- 1$ .

$\sqrt{16 a^{4} + 8 a^{2} b^{2} + b^{4} - 48 a^{2} + 24 a b - 24 b a + 12 b^{2}}$

Step 4.2

Subtract $24 b a$ from $24 a b$ .

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Step 4.2.1

Move $b$ .

$\sqrt{16 a^{4} + 8 a^{2} b^{2} + b^{4} - 48 a^{2} + 24 a b - 24 a b + 12 b^{2}}$

Step 4.2.2

Subtract $24 a b$ from $24 a b$ .

$\sqrt{16 a^{4} + 8 a^{2} b^{2} + b^{4} - 48 a^{2} + 0 + 12 b^{2}}$

Step 4.3

Add $- 48 a^{2}$ and $0$ .

$\sqrt{16 a^{4} + 8 a^{2} b^{2} + b^{4} - 48 a^{2} + 12 b^{2}}$