Algebra Examples

${(2 x + 1)}^{3} {(2 x - 1)}^{3}$

Step 1

Use the Binomial Theorem.

$({(2 x)}^{3} + 3 {(2 x)}^{2} \cdot 1 + 3 (2 x) \cdot 1^{2} + 1^{3}) {(2 x - 1)}^{3}$

Step 2

Simplify each term.

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Step 2.1

Apply the product rule to $2 x$ .

$(2^{3} x^{3} + 3 {(2 x)}^{2} \cdot 1 + 3 (2 x) \cdot 1^{2} + 1^{3}) {(2 x - 1)}^{3}$

Step 2.2

Raise $2$ to the power of $3$ .

$(8 x^{3} + 3 {(2 x)}^{2} \cdot 1 + 3 (2 x) \cdot 1^{2} + 1^{3}) {(2 x - 1)}^{3}$

Step 2.3

Apply the product rule to $2 x$ .

$(8 x^{3} + 3 (2^{2} x^{2}) \cdot 1 + 3 (2 x) \cdot 1^{2} + 1^{3}) {(2 x - 1)}^{3}$

Step 2.4

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

$(8 x^{3} + 3 (4 x^{2}) \cdot 1 + 3 (2 x) \cdot 1^{2} + 1^{3}) {(2 x - 1)}^{3}$

Step 2.5

Multiply $4$ by $3$ .

$(8 x^{3} + 12 x^{2} \cdot 1 + 3 (2 x) \cdot 1^{2} + 1^{3}) {(2 x - 1)}^{3}$

Step 2.6

Multiply $12$ by $1$ .

$(8 x^{3} + 12 x^{2} + 3 (2 x) \cdot 1^{2} + 1^{3}) {(2 x - 1)}^{3}$

Step 2.7

Multiply $2$ by $3$ .

$(8 x^{3} + 12 x^{2} + 6 x \cdot 1^{2} + 1^{3}) {(2 x - 1)}^{3}$

Step 2.8

One to any power is one.

$(8 x^{3} + 12 x^{2} + 6 x \cdot 1 + 1^{3}) {(2 x - 1)}^{3}$

Step 2.9

Multiply $6$ by $1$ .

$(8 x^{3} + 12 x^{2} + 6 x + 1^{3}) {(2 x - 1)}^{3}$

Step 2.10

One to any power is one.

$(8 x^{3} + 12 x^{2} + 6 x + 1) {(2 x - 1)}^{3}$

Step 3

Use the Binomial Theorem.

$(8 x^{3} + 12 x^{2} + 6 x + 1) ({(2 x)}^{3} + 3 {(2 x)}^{2} \cdot - 1 + 3 (2 x) {(- 1)}^{2} + {(- 1)}^{3})$

Step 4

Simplify each term.

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Step 4.1

Apply the product rule to $2 x$ .

$(8 x^{3} + 12 x^{2} + 6 x + 1) (2^{3} x^{3} + 3 {(2 x)}^{2} \cdot - 1 + 3 (2 x) {(- 1)}^{2} + {(- 1)}^{3})$

Step 4.2

Raise $2$ to the power of $3$ .

$(8 x^{3} + 12 x^{2} + 6 x + 1) (8 x^{3} + 3 {(2 x)}^{2} \cdot - 1 + 3 (2 x) {(- 1)}^{2} + {(- 1)}^{3})$

Step 4.3

Apply the product rule to $2 x$ .

$(8 x^{3} + 12 x^{2} + 6 x + 1) (8 x^{3} + 3 (2^{2} x^{2}) \cdot - 1 + 3 (2 x) {(- 1)}^{2} + {(- 1)}^{3})$

Step 4.4

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

$(8 x^{3} + 12 x^{2} + 6 x + 1) (8 x^{3} + 3 (4 x^{2}) \cdot - 1 + 3 (2 x) {(- 1)}^{2} + {(- 1)}^{3})$

Step 4.5

Multiply $4$ by $3$ .

$(8 x^{3} + 12 x^{2} + 6 x + 1) (8 x^{3} + 12 x^{2} \cdot - 1 + 3 (2 x) {(- 1)}^{2} + {(- 1)}^{3})$

Step 4.6

Multiply $- 1$ by $12$ .

$(8 x^{3} + 12 x^{2} + 6 x + 1) (8 x^{3} - 12 x^{2} + 3 (2 x) {(- 1)}^{2} + {(- 1)}^{3})$

Step 4.7

Multiply $2$ by $3$ .

$(8 x^{3} + 12 x^{2} + 6 x + 1) (8 x^{3} - 12 x^{2} + 6 x {(- 1)}^{2} + {(- 1)}^{3})$

Step 4.8

Raise $- 1$ to the power of $2$ .

$(8 x^{3} + 12 x^{2} + 6 x + 1) (8 x^{3} - 12 x^{2} + 6 x \cdot 1 + {(- 1)}^{3})$

Step 4.9

Multiply $6$ by $1$ .

$(8 x^{3} + 12 x^{2} + 6 x + 1) (8 x^{3} - 12 x^{2} + 6 x + {(- 1)}^{3})$

Step 4.10

Raise $- 1$ to the power of $3$ .

$(8 x^{3} + 12 x^{2} + 6 x + 1) (8 x^{3} - 12 x^{2} + 6 x - 1)$

Step 5

Expand $(8 x^{3} + 12 x^{2} + 6 x + 1) (8 x^{3} - 12 x^{2} + 6 x - 1)$ by multiplying each term in the first expression by each term in the second expression.

$8 x^{3} (8 x^{3}) + 8 x^{3} (- 12 x^{2}) + 8 x^{3} (6 x) + 8 x^{3} \cdot - 1 + 12 x^{2} (8 x^{3}) + 12 x^{2} (- 12 x^{2}) + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 6 x \cdot - 1 + 1 (8 x^{3}) + 1 (- 12 x^{2}) + 1 (6 x) + 1 \cdot - 1$

Step 6

Simplify terms.

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Step 6.1

Combine the opposite terms in $8 x^{3} (8 x^{3}) + 8 x^{3} (- 12 x^{2}) + 8 x^{3} (6 x) + 8 x^{3} \cdot - 1 + 12 x^{2} (8 x^{3}) + 12 x^{2} (- 12 x^{2}) + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 6 x \cdot - 1 + 1 (8 x^{3}) + 1 (- 12 x^{2}) + 1 (6 x) + 1 \cdot - 1$ .

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Step 6.1.1

Reorder the factors in the terms $8 x^{3} \cdot - 1$ and $1 (8 x^{3})$ .

$8 x^{3} (8 x^{3}) + 8 x^{3} (- 12 x^{2}) + 8 x^{3} (6 x) - 1 \cdot 8 x^{3} + 12 x^{2} (8 x^{3}) + 12 x^{2} (- 12 x^{2}) + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 6 x \cdot - 1 + 1 \cdot 8 x^{3} + 1 (- 12 x^{2}) + 1 (6 x) + 1 \cdot - 1$

Step 6.1.2

Add $- 1 \cdot 8 x^{3}$ and $1 \cdot 8 x^{3}$ .

$8 x^{3} (8 x^{3}) + 8 x^{3} (- 12 x^{2}) + 8 x^{3} (6 x) + 12 x^{2} (8 x^{3}) + 12 x^{2} (- 12 x^{2}) + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 6 x \cdot - 1 + 0 + 1 (- 12 x^{2}) + 1 (6 x) + 1 \cdot - 1$

Step 6.1.3

Add $8 x^{3} (8 x^{3}) + 8 x^{3} (- 12 x^{2}) + 8 x^{3} (6 x) + 12 x^{2} (8 x^{3}) + 12 x^{2} (- 12 x^{2}) + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 6 x \cdot - 1$ and $0$ .

$8 x^{3} (8 x^{3}) + 8 x^{3} (- 12 x^{2}) + 8 x^{3} (6 x) + 12 x^{2} (8 x^{3}) + 12 x^{2} (- 12 x^{2}) + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 6 x \cdot - 1 + 1 (- 12 x^{2}) + 1 (6 x) + 1 \cdot - 1$

Step 6.1.4

Reorder the factors in the terms $6 x \cdot - 1$ and $1 (6 x)$ .

$8 x^{3} (8 x^{3}) + 8 x^{3} (- 12 x^{2}) + 8 x^{3} (6 x) + 12 x^{2} (8 x^{3}) + 12 x^{2} (- 12 x^{2}) + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) - 1 \cdot 6 x + 1 (- 12 x^{2}) + 1 \cdot 6 x + 1 \cdot - 1$

Step 6.1.5

Add $- 1 \cdot 6 x$ and $1 \cdot 6 x$ .

$8 x^{3} (8 x^{3}) + 8 x^{3} (- 12 x^{2}) + 8 x^{3} (6 x) + 12 x^{2} (8 x^{3}) + 12 x^{2} (- 12 x^{2}) + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 0 + 1 \cdot - 1$

Step 6.1.6

Add $8 x^{3} (8 x^{3}) + 8 x^{3} (- 12 x^{2}) + 8 x^{3} (6 x) + 12 x^{2} (8 x^{3}) + 12 x^{2} (- 12 x^{2}) + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2})$ and $0$ .

$8 x^{3} (8 x^{3}) + 8 x^{3} (- 12 x^{2}) + 8 x^{3} (6 x) + 12 x^{2} (8 x^{3}) + 12 x^{2} (- 12 x^{2}) + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2

Simplify each term.

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Step 6.2.1

Rewrite using the commutative property of multiplication.

$8 \cdot 8 x^{3} x^{3} + 8 x^{3} (- 12 x^{2}) + 8 x^{3} (6 x) + 12 x^{2} (8 x^{3}) + 12 x^{2} (- 12 x^{2}) + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.2

Multiply $x^{3}$ by $x^{3}$ by adding the exponents.

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Step 6.2.2.1

Move $x^{3}$ .

$8 \cdot 8 (x^{3} x^{3}) + 8 x^{3} (- 12 x^{2}) + 8 x^{3} (6 x) + 12 x^{2} (8 x^{3}) + 12 x^{2} (- 12 x^{2}) + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.2.2

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

$8 \cdot 8 x^{3 + 3} + 8 x^{3} (- 12 x^{2}) + 8 x^{3} (6 x) + 12 x^{2} (8 x^{3}) + 12 x^{2} (- 12 x^{2}) + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.2.3

Add $3$ and $3$ .

$8 \cdot 8 x^{6} + 8 x^{3} (- 12 x^{2}) + 8 x^{3} (6 x) + 12 x^{2} (8 x^{3}) + 12 x^{2} (- 12 x^{2}) + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.3

Multiply $8$ by $8$ .

$64 x^{6} + 8 x^{3} (- 12 x^{2}) + 8 x^{3} (6 x) + 12 x^{2} (8 x^{3}) + 12 x^{2} (- 12 x^{2}) + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.4

Rewrite using the commutative property of multiplication.

$64 x^{6} + 8 \cdot - 12 x^{3} x^{2} + 8 x^{3} (6 x) + 12 x^{2} (8 x^{3}) + 12 x^{2} (- 12 x^{2}) + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.5

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 6.2.5.1

Move $x^{2}$ .

$64 x^{6} + 8 \cdot - 12 (x^{2} x^{3}) + 8 x^{3} (6 x) + 12 x^{2} (8 x^{3}) + 12 x^{2} (- 12 x^{2}) + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.5.2

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

$64 x^{6} + 8 \cdot - 12 x^{2 + 3} + 8 x^{3} (6 x) + 12 x^{2} (8 x^{3}) + 12 x^{2} (- 12 x^{2}) + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.5.3

Add $2$ and $3$ .

$64 x^{6} + 8 \cdot - 12 x^{5} + 8 x^{3} (6 x) + 12 x^{2} (8 x^{3}) + 12 x^{2} (- 12 x^{2}) + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.6

Multiply $8$ by $- 12$ .

$64 x^{6} - 96 x^{5} + 8 x^{3} (6 x) + 12 x^{2} (8 x^{3}) + 12 x^{2} (- 12 x^{2}) + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.7

Rewrite using the commutative property of multiplication.

$64 x^{6} - 96 x^{5} + 8 \cdot 6 x^{3} x + 12 x^{2} (8 x^{3}) + 12 x^{2} (- 12 x^{2}) + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.8

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 6.2.8.1

Move $x$ .

$64 x^{6} - 96 x^{5} + 8 \cdot 6 (x \cdot x^{3}) + 12 x^{2} (8 x^{3}) + 12 x^{2} (- 12 x^{2}) + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.8.2

Multiply $x$ by $x^{3}$ .

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Step 6.2.8.2.1

Raise $x$ to the power of $1$ .

$64 x^{6} - 96 x^{5} + 8 \cdot 6 (x^{1} x^{3}) + 12 x^{2} (8 x^{3}) + 12 x^{2} (- 12 x^{2}) + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.8.2.2

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

$64 x^{6} - 96 x^{5} + 8 \cdot 6 x^{1 + 3} + 12 x^{2} (8 x^{3}) + 12 x^{2} (- 12 x^{2}) + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.8.3

Add $1$ and $3$ .

$64 x^{6} - 96 x^{5} + 8 \cdot 6 x^{4} + 12 x^{2} (8 x^{3}) + 12 x^{2} (- 12 x^{2}) + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.9

Multiply $8$ by $6$ .

$64 x^{6} - 96 x^{5} + 48 x^{4} + 12 x^{2} (8 x^{3}) + 12 x^{2} (- 12 x^{2}) + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.10

Rewrite using the commutative property of multiplication.

$64 x^{6} - 96 x^{5} + 48 x^{4} + 12 \cdot 8 x^{2} x^{3} + 12 x^{2} (- 12 x^{2}) + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.11

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 6.2.11.1

Move $x^{3}$ .

$64 x^{6} - 96 x^{5} + 48 x^{4} + 12 \cdot 8 (x^{3} x^{2}) + 12 x^{2} (- 12 x^{2}) + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.11.2

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

$64 x^{6} - 96 x^{5} + 48 x^{4} + 12 \cdot 8 x^{3 + 2} + 12 x^{2} (- 12 x^{2}) + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.11.3

Add $3$ and $2$ .

$64 x^{6} - 96 x^{5} + 48 x^{4} + 12 \cdot 8 x^{5} + 12 x^{2} (- 12 x^{2}) + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.12

Multiply $12$ by $8$ .

$64 x^{6} - 96 x^{5} + 48 x^{4} + 96 x^{5} + 12 x^{2} (- 12 x^{2}) + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.13

Rewrite using the commutative property of multiplication.

$64 x^{6} - 96 x^{5} + 48 x^{4} + 96 x^{5} + 12 \cdot - 12 x^{2} x^{2} + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.14

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

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Step 6.2.14.1

Move $x^{2}$ .

$64 x^{6} - 96 x^{5} + 48 x^{4} + 96 x^{5} + 12 \cdot - 12 (x^{2} x^{2}) + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.14.2

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

$64 x^{6} - 96 x^{5} + 48 x^{4} + 96 x^{5} + 12 \cdot - 12 x^{2 + 2} + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.14.3

Add $2$ and $2$ .

$64 x^{6} - 96 x^{5} + 48 x^{4} + 96 x^{5} + 12 \cdot - 12 x^{4} + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.15

Multiply $12$ by $- 12$ .

$64 x^{6} - 96 x^{5} + 48 x^{4} + 96 x^{5} - 144 x^{4} + 12 x^{2} (6 x) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.16

Rewrite using the commutative property of multiplication.

$64 x^{6} - 96 x^{5} + 48 x^{4} + 96 x^{5} - 144 x^{4} + 12 \cdot 6 x^{2} x + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.17

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

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Step 6.2.17.1

Move $x$ .

$64 x^{6} - 96 x^{5} + 48 x^{4} + 96 x^{5} - 144 x^{4} + 12 \cdot 6 (x \cdot x^{2}) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.17.2

Multiply $x$ by $x^{2}$ .

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Step 6.2.17.2.1

Raise $x$ to the power of $1$ .

$64 x^{6} - 96 x^{5} + 48 x^{4} + 96 x^{5} - 144 x^{4} + 12 \cdot 6 (x^{1} x^{2}) + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.17.2.2

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

$64 x^{6} - 96 x^{5} + 48 x^{4} + 96 x^{5} - 144 x^{4} + 12 \cdot 6 x^{1 + 2} + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.17.3

Add $1$ and $2$ .

$64 x^{6} - 96 x^{5} + 48 x^{4} + 96 x^{5} - 144 x^{4} + 12 \cdot 6 x^{3} + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.18

Multiply $12$ by $6$ .

$64 x^{6} - 96 x^{5} + 48 x^{4} + 96 x^{5} - 144 x^{4} + 72 x^{3} + 12 x^{2} \cdot - 1 + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.19

Multiply $- 1$ by $12$ .

$64 x^{6} - 96 x^{5} + 48 x^{4} + 96 x^{5} - 144 x^{4} + 72 x^{3} - 12 x^{2} + 6 x (8 x^{3}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.20

Rewrite using the commutative property of multiplication.

$64 x^{6} - 96 x^{5} + 48 x^{4} + 96 x^{5} - 144 x^{4} + 72 x^{3} - 12 x^{2} + 6 \cdot 8 x \cdot x^{3} + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.21

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 6.2.21.1

Move $x^{3}$ .

$64 x^{6} - 96 x^{5} + 48 x^{4} + 96 x^{5} - 144 x^{4} + 72 x^{3} - 12 x^{2} + 6 \cdot 8 (x^{3} x) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.21.2

Multiply $x^{3}$ by $x$ .

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Step 6.2.21.2.1

Raise $x$ to the power of $1$ .

$64 x^{6} - 96 x^{5} + 48 x^{4} + 96 x^{5} - 144 x^{4} + 72 x^{3} - 12 x^{2} + 6 \cdot 8 (x^{3} x^{1}) + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.21.2.2

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

$64 x^{6} - 96 x^{5} + 48 x^{4} + 96 x^{5} - 144 x^{4} + 72 x^{3} - 12 x^{2} + 6 \cdot 8 x^{3 + 1} + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.21.3

Add $3$ and $1$ .

$64 x^{6} - 96 x^{5} + 48 x^{4} + 96 x^{5} - 144 x^{4} + 72 x^{3} - 12 x^{2} + 6 \cdot 8 x^{4} + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.22

Multiply $6$ by $8$ .

$64 x^{6} - 96 x^{5} + 48 x^{4} + 96 x^{5} - 144 x^{4} + 72 x^{3} - 12 x^{2} + 48 x^{4} + 6 x (- 12 x^{2}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.23

Rewrite using the commutative property of multiplication.

$64 x^{6} - 96 x^{5} + 48 x^{4} + 96 x^{5} - 144 x^{4} + 72 x^{3} - 12 x^{2} + 48 x^{4} + 6 \cdot - 12 x \cdot x^{2} + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.24

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

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Step 6.2.24.1

Move $x^{2}$ .

$64 x^{6} - 96 x^{5} + 48 x^{4} + 96 x^{5} - 144 x^{4} + 72 x^{3} - 12 x^{2} + 48 x^{4} + 6 \cdot - 12 (x^{2} x) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.24.2

Multiply $x^{2}$ by $x$ .

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Step 6.2.24.2.1

Raise $x$ to the power of $1$ .

$64 x^{6} - 96 x^{5} + 48 x^{4} + 96 x^{5} - 144 x^{4} + 72 x^{3} - 12 x^{2} + 48 x^{4} + 6 \cdot - 12 (x^{2} x^{1}) + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.24.2.2

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

$64 x^{6} - 96 x^{5} + 48 x^{4} + 96 x^{5} - 144 x^{4} + 72 x^{3} - 12 x^{2} + 48 x^{4} + 6 \cdot - 12 x^{2 + 1} + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.24.3

Add $2$ and $1$ .

$64 x^{6} - 96 x^{5} + 48 x^{4} + 96 x^{5} - 144 x^{4} + 72 x^{3} - 12 x^{2} + 48 x^{4} + 6 \cdot - 12 x^{3} + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.25

Multiply $6$ by $- 12$ .

$64 x^{6} - 96 x^{5} + 48 x^{4} + 96 x^{5} - 144 x^{4} + 72 x^{3} - 12 x^{2} + 48 x^{4} - 72 x^{3} + 6 x (6 x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.26

Rewrite using the commutative property of multiplication.

$64 x^{6} - 96 x^{5} + 48 x^{4} + 96 x^{5} - 144 x^{4} + 72 x^{3} - 12 x^{2} + 48 x^{4} - 72 x^{3} + 6 \cdot 6 x \cdot x + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.27

Multiply $x$ by $x$ by adding the exponents.

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Step 6.2.27.1

Move $x$ .

$64 x^{6} - 96 x^{5} + 48 x^{4} + 96 x^{5} - 144 x^{4} + 72 x^{3} - 12 x^{2} + 48 x^{4} - 72 x^{3} + 6 \cdot 6 (x \cdot x) + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.27.2

Multiply $x$ by $x$ .

$64 x^{6} - 96 x^{5} + 48 x^{4} + 96 x^{5} - 144 x^{4} + 72 x^{3} - 12 x^{2} + 48 x^{4} - 72 x^{3} + 6 \cdot 6 x^{2} + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.28

Multiply $6$ by $6$ .

$64 x^{6} - 96 x^{5} + 48 x^{4} + 96 x^{5} - 144 x^{4} + 72 x^{3} - 12 x^{2} + 48 x^{4} - 72 x^{3} + 36 x^{2} + 1 (- 12 x^{2}) + 1 \cdot - 1$

Step 6.2.29

Multiply $- 12 x^{2}$ by $1$ .

$64 x^{6} - 96 x^{5} + 48 x^{4} + 96 x^{5} - 144 x^{4} + 72 x^{3} - 12 x^{2} + 48 x^{4} - 72 x^{3} + 36 x^{2} - 12 x^{2} + 1 \cdot - 1$

Step 6.2.30

Multiply $- 1$ by $1$ .

$64 x^{6} - 96 x^{5} + 48 x^{4} + 96 x^{5} - 144 x^{4} + 72 x^{3} - 12 x^{2} + 48 x^{4} - 72 x^{3} + 36 x^{2} - 12 x^{2} - 1$

Step 6.3

Simplify by adding terms.

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Step 6.3.1

Combine the opposite terms in $64 x^{6} - 96 x^{5} + 48 x^{4} + 96 x^{5} - 144 x^{4} + 72 x^{3} - 12 x^{2} + 48 x^{4} - 72 x^{3} + 36 x^{2} - 12 x^{2} - 1$ .

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Step 6.3.1.1

Add $- 96 x^{5}$ and $96 x^{5}$ .

$64 x^{6} + 48 x^{4} + 0 - 144 x^{4} + 72 x^{3} - 12 x^{2} + 48 x^{4} - 72 x^{3} + 36 x^{2} - 12 x^{2} - 1$

Step 6.3.1.2

Add $64 x^{6} + 48 x^{4}$ and $0$ .

$64 x^{6} + 48 x^{4} - 144 x^{4} + 72 x^{3} - 12 x^{2} + 48 x^{4} - 72 x^{3} + 36 x^{2} - 12 x^{2} - 1$

Step 6.3.1.3

Subtract $72 x^{3}$ from $72 x^{3}$ .

$64 x^{6} + 48 x^{4} - 144 x^{4} - 12 x^{2} + 48 x^{4} + 0 + 36 x^{2} - 12 x^{2} - 1$

Step 6.3.1.4

Add $64 x^{6} + 48 x^{4} - 144 x^{4} - 12 x^{2} + 48 x^{4}$ and $0$ .

$64 x^{6} + 48 x^{4} - 144 x^{4} - 12 x^{2} + 48 x^{4} + 36 x^{2} - 12 x^{2} - 1$

Step 6.3.2

Subtract $144 x^{4}$ from $48 x^{4}$ .

$64 x^{6} - 96 x^{4} - 12 x^{2} + 48 x^{4} + 36 x^{2} - 12 x^{2} - 1$

Step 6.3.3

Add $- 96 x^{4}$ and $48 x^{4}$ .

$64 x^{6} - 48 x^{4} - 12 x^{2} + 36 x^{2} - 12 x^{2} - 1$

Step 6.3.4

Add $- 12 x^{2}$ and $36 x^{2}$ .

$64 x^{6} - 48 x^{4} + 24 x^{2} - 12 x^{2} - 1$

Step 6.3.5

Subtract $12 x^{2}$ from $24 x^{2}$ .

$64 x^{6} - 48 x^{4} + 12 x^{2} - 1$