Basic Math Examples

${(a - 6 b)}^{2} {(a + 6 b)}^{2}$

Step 1

Rewrite ${(a - 6 b)}^{2}$ as $(a - 6 b) (a - 6 b)$ .

$(a - 6 b) (a - 6 b) {(a + 6 b)}^{2}$

Step 2

Expand $(a - 6 b) (a - 6 b)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$(a (a - 6 b) - 6 b (a - 6 b)) {(a + 6 b)}^{2}$

Step 2.2

Apply the distributive property.

$(a \cdot a + a (- 6 b) - 6 b (a - 6 b)) {(a + 6 b)}^{2}$

Step 2.3

Apply the distributive property.

$(a \cdot a + a (- 6 b) - 6 b a - 6 b (- 6 b)) {(a + 6 b)}^{2}$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $a$ by $a$ .

$(a^{2} + a (- 6 b) - 6 b a - 6 b (- 6 b)) {(a + 6 b)}^{2}$

Step 3.1.2

Rewrite using the commutative property of multiplication.

$(a^{2} - 6 a b - 6 b a - 6 b (- 6 b)) {(a + 6 b)}^{2}$

Step 3.1.3

Rewrite using the commutative property of multiplication.

$(a^{2} - 6 a b - 6 b a - 6 \cdot - 6 b \cdot b) {(a + 6 b)}^{2}$

Step 3.1.4

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

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Step 3.1.4.1

Move $b$ .

$(a^{2} - 6 a b - 6 b a - 6 \cdot - 6 (b \cdot b)) {(a + 6 b)}^{2}$

Step 3.1.4.2

Multiply $b$ by $b$ .

$(a^{2} - 6 a b - 6 b a - 6 \cdot - 6 b^{2}) {(a + 6 b)}^{2}$

Step 3.1.5

Multiply $- 6$ by $- 6$ .

$(a^{2} - 6 a b - 6 b a + 36 b^{2}) {(a + 6 b)}^{2}$

Step 3.2

Subtract $6 b a$ from $- 6 a b$ .

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Step 3.2.1

Move $b$ .

$(a^{2} - 6 a b - 6 a b + 36 b^{2}) {(a + 6 b)}^{2}$

Step 3.2.2

Subtract $6 a b$ from $- 6 a b$ .

$(a^{2} - 12 a b + 36 b^{2}) {(a + 6 b)}^{2}$

Step 4

Rewrite ${(a + 6 b)}^{2}$ as $(a + 6 b) (a + 6 b)$ .

$(a^{2} - 12 a b + 36 b^{2}) ((a + 6 b) (a + 6 b))$

Step 5

Expand $(a + 6 b) (a + 6 b)$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$(a^{2} - 12 a b + 36 b^{2}) (a (a + 6 b) + 6 b (a + 6 b))$

Step 5.2

Apply the distributive property.

$(a^{2} - 12 a b + 36 b^{2}) (a \cdot a + a (6 b) + 6 b (a + 6 b))$

Step 5.3

Apply the distributive property.

$(a^{2} - 12 a b + 36 b^{2}) (a \cdot a + a (6 b) + 6 b a + 6 b (6 b))$

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $a$ by $a$ .

$(a^{2} - 12 a b + 36 b^{2}) (a^{2} + a (6 b) + 6 b a + 6 b (6 b))$

Step 6.1.2

Rewrite using the commutative property of multiplication.

$(a^{2} - 12 a b + 36 b^{2}) (a^{2} + 6 a b + 6 b a + 6 b (6 b))$

Step 6.1.3

Rewrite using the commutative property of multiplication.

$(a^{2} - 12 a b + 36 b^{2}) (a^{2} + 6 a b + 6 b a + 6 \cdot 6 b \cdot b)$

Step 6.1.4

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

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Step 6.1.4.1

Move $b$ .

$(a^{2} - 12 a b + 36 b^{2}) (a^{2} + 6 a b + 6 b a + 6 \cdot 6 (b \cdot b))$

Step 6.1.4.2

Multiply $b$ by $b$ .

$(a^{2} - 12 a b + 36 b^{2}) (a^{2} + 6 a b + 6 b a + 6 \cdot 6 b^{2})$

Step 6.1.5

Multiply $6$ by $6$ .

$(a^{2} - 12 a b + 36 b^{2}) (a^{2} + 6 a b + 6 b a + 36 b^{2})$

Step 6.2

Add $6 a b$ and $6 b a$ .

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Step 6.2.1

Move $b$ .

$(a^{2} - 12 a b + 36 b^{2}) (a^{2} + 6 a b + 6 a b + 36 b^{2})$

Step 6.2.2

Add $6 a b$ and $6 a b$ .

$(a^{2} - 12 a b + 36 b^{2}) (a^{2} + 12 a b + 36 b^{2})$

Step 7

Expand $(a^{2} - 12 a b + 36 b^{2}) (a^{2} + 12 a b + 36 b^{2})$ by multiplying each term in the first expression by each term in the second expression.

$a^{2} a^{2} + a^{2} (12 a b) + a^{2} (36 b^{2}) - 12 a b a^{2} - 12 a b (12 a b) - 12 a b (36 b^{2}) + 36 b^{2} a^{2} + 36 b^{2} (12 a b) + 36 b^{2} (36 b^{2})$

Step 8

Simplify terms.

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Step 8.1

Combine the opposite terms in $a^{2} a^{2} + a^{2} (12 a b) + a^{2} (36 b^{2}) - 12 a b a^{2} - 12 a b (12 a b) - 12 a b (36 b^{2}) + 36 b^{2} a^{2} + 36 b^{2} (12 a b) + 36 b^{2} (36 b^{2})$ .

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Step 8.1.1

Reorder the factors in the terms $a^{2} (12 a b)$ and $- 12 a b a^{2}$ .

$a^{2} a^{2} + 12 a^{2} a b + a^{2} (36 b^{2}) - 12 a^{2} a b - 12 a b (12 a b) - 12 a b (36 b^{2}) + 36 b^{2} a^{2} + 36 b^{2} (12 a b) + 36 b^{2} (36 b^{2})$

Step 8.1.2

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

$a^{2} a^{2} + a^{2} (36 b^{2}) + 0 - 12 a b (12 a b) - 12 a b (36 b^{2}) + 36 b^{2} a^{2} + 36 b^{2} (12 a b) + 36 b^{2} (36 b^{2})$

Step 8.1.3

Add $a^{2} a^{2} + a^{2} (36 b^{2})$ and $0$ .

$a^{2} a^{2} + a^{2} (36 b^{2}) - 12 a b (12 a b) - 12 a b (36 b^{2}) + 36 b^{2} a^{2} + 36 b^{2} (12 a b) + 36 b^{2} (36 b^{2})$

Step 8.1.4

Reorder the factors in the terms $- 12 a b (36 b^{2})$ and $36 b^{2} (12 a b)$ .

$a^{2} a^{2} + a^{2} (36 b^{2}) - 12 a b (12 a b) - 12 \cdot 36 b^{2} a b + 36 b^{2} a^{2} + 12 \cdot 36 b^{2} a b + 36 b^{2} (36 b^{2})$

Step 8.1.5

Add $- 12 \cdot 36 b^{2} a b$ and $12 \cdot 36 b^{2} a b$ .

$a^{2} a^{2} + a^{2} (36 b^{2}) - 12 a b (12 a b) + 36 b^{2} a^{2} + 0 + 36 b^{2} (36 b^{2})$

Step 8.1.6

Add $a^{2} a^{2} + a^{2} (36 b^{2}) - 12 a b (12 a b) + 36 b^{2} a^{2}$ and $0$ .

$a^{2} a^{2} + a^{2} (36 b^{2}) - 12 a b (12 a b) + 36 b^{2} a^{2} + 36 b^{2} (36 b^{2})$

Step 8.2

Simplify each term.

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Step 8.2.1

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

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Step 8.2.1.1

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

$a^{2 + 2} + a^{2} (36 b^{2}) - 12 a b (12 a b) + 36 b^{2} a^{2} + 36 b^{2} (36 b^{2})$

Step 8.2.1.2

Add $2$ and $2$ .

$a^{4} + a^{2} (36 b^{2}) - 12 a b (12 a b) + 36 b^{2} a^{2} + 36 b^{2} (36 b^{2})$

Step 8.2.2

Rewrite using the commutative property of multiplication.

$a^{4} + 36 a^{2} b^{2} - 12 a b (12 a b) + 36 b^{2} a^{2} + 36 b^{2} (36 b^{2})$

Step 8.2.3

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

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Step 8.2.3.1

Move $a$ .

$a^{4} + 36 a^{2} b^{2} - 12 (a \cdot a) b (12 b) + 36 b^{2} a^{2} + 36 b^{2} (36 b^{2})$

Step 8.2.3.2

Multiply $a$ by $a$ .

$a^{4} + 36 a^{2} b^{2} - 12 a^{2} b (12 b) + 36 b^{2} a^{2} + 36 b^{2} (36 b^{2})$

Step 8.2.4

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

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Step 8.2.4.1

Move $b$ .

$a^{4} + 36 a^{2} b^{2} - 12 a^{2} (b \cdot b) \cdot 12 + 36 b^{2} a^{2} + 36 b^{2} (36 b^{2})$

Step 8.2.4.2

Multiply $b$ by $b$ .

$a^{4} + 36 a^{2} b^{2} - 12 a^{2} b^{2} \cdot 12 + 36 b^{2} a^{2} + 36 b^{2} (36 b^{2})$

Step 8.2.5

Multiply $12$ by $- 12$ .

$a^{4} + 36 a^{2} b^{2} - 144 a^{2} b^{2} + 36 b^{2} a^{2} + 36 b^{2} (36 b^{2})$

Step 8.2.6

Rewrite using the commutative property of multiplication.

$a^{4} + 36 a^{2} b^{2} - 144 a^{2} b^{2} + 36 b^{2} a^{2} + 36 \cdot 36 b^{2} b^{2}$

Step 8.2.7

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

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Step 8.2.7.1

Move $b^{2}$ .

$a^{4} + 36 a^{2} b^{2} - 144 a^{2} b^{2} + 36 b^{2} a^{2} + 36 \cdot 36 (b^{2} b^{2})$

Step 8.2.7.2

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

$a^{4} + 36 a^{2} b^{2} - 144 a^{2} b^{2} + 36 b^{2} a^{2} + 36 \cdot 36 b^{2 + 2}$

Step 8.2.7.3

Add $2$ and $2$ .

$a^{4} + 36 a^{2} b^{2} - 144 a^{2} b^{2} + 36 b^{2} a^{2} + 36 \cdot 36 b^{4}$

Step 8.2.8

Multiply $36$ by $36$ .

$a^{4} + 36 a^{2} b^{2} - 144 a^{2} b^{2} + 36 b^{2} a^{2} + 1296 b^{4}$

Step 8.3

Subtract $144 a^{2} b^{2}$ from $36 a^{2} b^{2}$ .

$a^{4} - 108 a^{2} b^{2} + 36 b^{2} a^{2} + 1296 b^{4}$

Step 9

Add $- 108 a^{2} b^{2}$ and $36 b^{2} a^{2}$ .

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Step 9.1

Move $b^{2}$ .

$a^{4} - 108 a^{2} b^{2} + 36 a^{2} b^{2} + 1296 b^{4}$

Step 9.2

Add $- 108 a^{2} b^{2}$ and $36 a^{2} b^{2}$ .

$a^{4} - 72 a^{2} b^{2} + 1296 b^{4}$