Pre-Algebra Examples

$(8 u^{2} + 6 u + 3) (4 u^{2} - 7 u + 8) (8 u^{2} + 2 u - 3)$

Step 1

Expand $(8 u^{2} + 6 u + 3) (4 u^{2} - 7 u + 8)$ by multiplying each term in the first expression by each term in the second expression.

$(8 u^{2} (4 u^{2}) + 8 u^{2} (- 7 u) + 8 u^{2} \cdot 8 + 6 u (4 u^{2}) + 6 u (- 7 u) + 6 u \cdot 8 + 3 (4 u^{2}) + 3 (- 7 u) + 3 \cdot 8) (8 u^{2} + 2 u - 3)$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite using the commutative property of multiplication.

$(8 \cdot 4 u^{2} u^{2} + 8 u^{2} (- 7 u) + 8 u^{2} \cdot 8 + 6 u (4 u^{2}) + 6 u (- 7 u) + 6 u \cdot 8 + 3 (4 u^{2}) + 3 (- 7 u) + 3 \cdot 8) (8 u^{2} + 2 u - 3)$

Step 2.1.2

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

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Step 2.1.2.1

Move $u^{2}$ .

$(8 \cdot 4 (u^{2} u^{2}) + 8 u^{2} (- 7 u) + 8 u^{2} \cdot 8 + 6 u (4 u^{2}) + 6 u (- 7 u) + 6 u \cdot 8 + 3 (4 u^{2}) + 3 (- 7 u) + 3 \cdot 8) (8 u^{2} + 2 u - 3)$

Step 2.1.2.2

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

$(8 \cdot 4 u^{2 + 2} + 8 u^{2} (- 7 u) + 8 u^{2} \cdot 8 + 6 u (4 u^{2}) + 6 u (- 7 u) + 6 u \cdot 8 + 3 (4 u^{2}) + 3 (- 7 u) + 3 \cdot 8) (8 u^{2} + 2 u - 3)$

Step 2.1.2.3

Add $2$ and $2$ .

$(8 \cdot 4 u^{4} + 8 u^{2} (- 7 u) + 8 u^{2} \cdot 8 + 6 u (4 u^{2}) + 6 u (- 7 u) + 6 u \cdot 8 + 3 (4 u^{2}) + 3 (- 7 u) + 3 \cdot 8) (8 u^{2} + 2 u - 3)$

Step 2.1.3

Multiply $8$ by $4$ .

$(32 u^{4} + 8 u^{2} (- 7 u) + 8 u^{2} \cdot 8 + 6 u (4 u^{2}) + 6 u (- 7 u) + 6 u \cdot 8 + 3 (4 u^{2}) + 3 (- 7 u) + 3 \cdot 8) (8 u^{2} + 2 u - 3)$

Step 2.1.4

Rewrite using the commutative property of multiplication.

$(32 u^{4} + 8 \cdot - 7 u^{2} u + 8 u^{2} \cdot 8 + 6 u (4 u^{2}) + 6 u (- 7 u) + 6 u \cdot 8 + 3 (4 u^{2}) + 3 (- 7 u) + 3 \cdot 8) (8 u^{2} + 2 u - 3)$

Step 2.1.5

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

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Step 2.1.5.1

Move $u$ .

$(32 u^{4} + 8 \cdot - 7 (u \cdot u^{2}) + 8 u^{2} \cdot 8 + 6 u (4 u^{2}) + 6 u (- 7 u) + 6 u \cdot 8 + 3 (4 u^{2}) + 3 (- 7 u) + 3 \cdot 8) (8 u^{2} + 2 u - 3)$

Step 2.1.5.2

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

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Step 2.1.5.2.1

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

$(32 u^{4} + 8 \cdot - 7 (u^{1} u^{2}) + 8 u^{2} \cdot 8 + 6 u (4 u^{2}) + 6 u (- 7 u) + 6 u \cdot 8 + 3 (4 u^{2}) + 3 (- 7 u) + 3 \cdot 8) (8 u^{2} + 2 u - 3)$

Step 2.1.5.2.2

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

$(32 u^{4} + 8 \cdot - 7 u^{1 + 2} + 8 u^{2} \cdot 8 + 6 u (4 u^{2}) + 6 u (- 7 u) + 6 u \cdot 8 + 3 (4 u^{2}) + 3 (- 7 u) + 3 \cdot 8) (8 u^{2} + 2 u - 3)$

Step 2.1.5.3

Add $1$ and $2$ .

$(32 u^{4} + 8 \cdot - 7 u^{3} + 8 u^{2} \cdot 8 + 6 u (4 u^{2}) + 6 u (- 7 u) + 6 u \cdot 8 + 3 (4 u^{2}) + 3 (- 7 u) + 3 \cdot 8) (8 u^{2} + 2 u - 3)$

Step 2.1.6

Multiply $8$ by $- 7$ .

$(32 u^{4} - 56 u^{3} + 8 u^{2} \cdot 8 + 6 u (4 u^{2}) + 6 u (- 7 u) + 6 u \cdot 8 + 3 (4 u^{2}) + 3 (- 7 u) + 3 \cdot 8) (8 u^{2} + 2 u - 3)$

Step 2.1.7

Multiply $8$ by $8$ .

$(32 u^{4} - 56 u^{3} + 64 u^{2} + 6 u (4 u^{2}) + 6 u (- 7 u) + 6 u \cdot 8 + 3 (4 u^{2}) + 3 (- 7 u) + 3 \cdot 8) (8 u^{2} + 2 u - 3)$

Step 2.1.8

Rewrite using the commutative property of multiplication.

$(32 u^{4} - 56 u^{3} + 64 u^{2} + 6 \cdot 4 u \cdot u^{2} + 6 u (- 7 u) + 6 u \cdot 8 + 3 (4 u^{2}) + 3 (- 7 u) + 3 \cdot 8) (8 u^{2} + 2 u - 3)$

Step 2.1.9

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

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Step 2.1.9.1

Move $u^{2}$ .

$(32 u^{4} - 56 u^{3} + 64 u^{2} + 6 \cdot 4 (u^{2} u) + 6 u (- 7 u) + 6 u \cdot 8 + 3 (4 u^{2}) + 3 (- 7 u) + 3 \cdot 8) (8 u^{2} + 2 u - 3)$

Step 2.1.9.2

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

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Step 2.1.9.2.1

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

$(32 u^{4} - 56 u^{3} + 64 u^{2} + 6 \cdot 4 (u^{2} u^{1}) + 6 u (- 7 u) + 6 u \cdot 8 + 3 (4 u^{2}) + 3 (- 7 u) + 3 \cdot 8) (8 u^{2} + 2 u - 3)$

Step 2.1.9.2.2

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

$(32 u^{4} - 56 u^{3} + 64 u^{2} + 6 \cdot 4 u^{2 + 1} + 6 u (- 7 u) + 6 u \cdot 8 + 3 (4 u^{2}) + 3 (- 7 u) + 3 \cdot 8) (8 u^{2} + 2 u - 3)$

Step 2.1.9.3

Add $2$ and $1$ .

$(32 u^{4} - 56 u^{3} + 64 u^{2} + 6 \cdot 4 u^{3} + 6 u (- 7 u) + 6 u \cdot 8 + 3 (4 u^{2}) + 3 (- 7 u) + 3 \cdot 8) (8 u^{2} + 2 u - 3)$

Step 2.1.10

Multiply $6$ by $4$ .

$(32 u^{4} - 56 u^{3} + 64 u^{2} + 24 u^{3} + 6 u (- 7 u) + 6 u \cdot 8 + 3 (4 u^{2}) + 3 (- 7 u) + 3 \cdot 8) (8 u^{2} + 2 u - 3)$

Step 2.1.11

Rewrite using the commutative property of multiplication.

$(32 u^{4} - 56 u^{3} + 64 u^{2} + 24 u^{3} + 6 \cdot - 7 u \cdot u + 6 u \cdot 8 + 3 (4 u^{2}) + 3 (- 7 u) + 3 \cdot 8) (8 u^{2} + 2 u - 3)$

Step 2.1.12

Multiply $u$ by $u$ by adding the exponents.

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Step 2.1.12.1

Move $u$ .

$(32 u^{4} - 56 u^{3} + 64 u^{2} + 24 u^{3} + 6 \cdot - 7 (u \cdot u) + 6 u \cdot 8 + 3 (4 u^{2}) + 3 (- 7 u) + 3 \cdot 8) (8 u^{2} + 2 u - 3)$

Step 2.1.12.2

Multiply $u$ by $u$ .

$(32 u^{4} - 56 u^{3} + 64 u^{2} + 24 u^{3} + 6 \cdot - 7 u^{2} + 6 u \cdot 8 + 3 (4 u^{2}) + 3 (- 7 u) + 3 \cdot 8) (8 u^{2} + 2 u - 3)$

Step 2.1.13

Multiply $6$ by $- 7$ .

$(32 u^{4} - 56 u^{3} + 64 u^{2} + 24 u^{3} - 42 u^{2} + 6 u \cdot 8 + 3 (4 u^{2}) + 3 (- 7 u) + 3 \cdot 8) (8 u^{2} + 2 u - 3)$

Step 2.1.14

Multiply $8$ by $6$ .

$(32 u^{4} - 56 u^{3} + 64 u^{2} + 24 u^{3} - 42 u^{2} + 48 u + 3 (4 u^{2}) + 3 (- 7 u) + 3 \cdot 8) (8 u^{2} + 2 u - 3)$

Step 2.1.15

Multiply $4$ by $3$ .

$(32 u^{4} - 56 u^{3} + 64 u^{2} + 24 u^{3} - 42 u^{2} + 48 u + 12 u^{2} + 3 (- 7 u) + 3 \cdot 8) (8 u^{2} + 2 u - 3)$

Step 2.1.16

Multiply $- 7$ by $3$ .

$(32 u^{4} - 56 u^{3} + 64 u^{2} + 24 u^{3} - 42 u^{2} + 48 u + 12 u^{2} - 21 u + 3 \cdot 8) (8 u^{2} + 2 u - 3)$

Step 2.1.17

Multiply $3$ by $8$ .

$(32 u^{4} - 56 u^{3} + 64 u^{2} + 24 u^{3} - 42 u^{2} + 48 u + 12 u^{2} - 21 u + 24) (8 u^{2} + 2 u - 3)$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Add $- 56 u^{3}$ and $24 u^{3}$ .

$(32 u^{4} - 32 u^{3} + 64 u^{2} - 42 u^{2} + 48 u + 12 u^{2} - 21 u + 24) (8 u^{2} + 2 u - 3)$

Step 2.2.2

Subtract $42 u^{2}$ from $64 u^{2}$ .

$(32 u^{4} - 32 u^{3} + 22 u^{2} + 48 u + 12 u^{2} - 21 u + 24) (8 u^{2} + 2 u - 3)$

Step 2.2.3

Add $22 u^{2}$ and $12 u^{2}$ .

$(32 u^{4} - 32 u^{3} + 34 u^{2} + 48 u - 21 u + 24) (8 u^{2} + 2 u - 3)$

Step 2.2.4

Subtract $21 u$ from $48 u$ .

$(32 u^{4} - 32 u^{3} + 34 u^{2} + 27 u + 24) (8 u^{2} + 2 u - 3)$

Step 3

Expand $(32 u^{4} - 32 u^{3} + 34 u^{2} + 27 u + 24) (8 u^{2} + 2 u - 3)$ by multiplying each term in the first expression by each term in the second expression.

$32 u^{4} (8 u^{2}) + 32 u^{4} (2 u) + 32 u^{4} \cdot - 3 - 32 u^{3} (8 u^{2}) - 32 u^{3} (2 u) - 32 u^{3} \cdot - 3 + 34 u^{2} (8 u^{2}) + 34 u^{2} (2 u) + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4

Simplify 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.

$32 \cdot 8 u^{4} u^{2} + 32 u^{4} (2 u) + 32 u^{4} \cdot - 3 - 32 u^{3} (8 u^{2}) - 32 u^{3} (2 u) - 32 u^{3} \cdot - 3 + 34 u^{2} (8 u^{2}) + 34 u^{2} (2 u) + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.2

Multiply $u^{4}$ by $u^{2}$ by adding the exponents.

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Step 4.1.2.1

Move $u^{2}$ .

$32 \cdot 8 (u^{2} u^{4}) + 32 u^{4} (2 u) + 32 u^{4} \cdot - 3 - 32 u^{3} (8 u^{2}) - 32 u^{3} (2 u) - 32 u^{3} \cdot - 3 + 34 u^{2} (8 u^{2}) + 34 u^{2} (2 u) + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.2.2

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

$32 \cdot 8 u^{2 + 4} + 32 u^{4} (2 u) + 32 u^{4} \cdot - 3 - 32 u^{3} (8 u^{2}) - 32 u^{3} (2 u) - 32 u^{3} \cdot - 3 + 34 u^{2} (8 u^{2}) + 34 u^{2} (2 u) + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.2.3

Add $2$ and $4$ .

$32 \cdot 8 u^{6} + 32 u^{4} (2 u) + 32 u^{4} \cdot - 3 - 32 u^{3} (8 u^{2}) - 32 u^{3} (2 u) - 32 u^{3} \cdot - 3 + 34 u^{2} (8 u^{2}) + 34 u^{2} (2 u) + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.3

Multiply $32$ by $8$ .

$256 u^{6} + 32 u^{4} (2 u) + 32 u^{4} \cdot - 3 - 32 u^{3} (8 u^{2}) - 32 u^{3} (2 u) - 32 u^{3} \cdot - 3 + 34 u^{2} (8 u^{2}) + 34 u^{2} (2 u) + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.4

Rewrite using the commutative property of multiplication.

$256 u^{6} + 32 \cdot 2 u^{4} u + 32 u^{4} \cdot - 3 - 32 u^{3} (8 u^{2}) - 32 u^{3} (2 u) - 32 u^{3} \cdot - 3 + 34 u^{2} (8 u^{2}) + 34 u^{2} (2 u) + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.5

Multiply $u^{4}$ by $u$ by adding the exponents.

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Step 4.1.5.1

Move $u$ .

$256 u^{6} + 32 \cdot 2 (u \cdot u^{4}) + 32 u^{4} \cdot - 3 - 32 u^{3} (8 u^{2}) - 32 u^{3} (2 u) - 32 u^{3} \cdot - 3 + 34 u^{2} (8 u^{2}) + 34 u^{2} (2 u) + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.5.2

Multiply $u$ by $u^{4}$ .

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Step 4.1.5.2.1

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

$256 u^{6} + 32 \cdot 2 (u^{1} u^{4}) + 32 u^{4} \cdot - 3 - 32 u^{3} (8 u^{2}) - 32 u^{3} (2 u) - 32 u^{3} \cdot - 3 + 34 u^{2} (8 u^{2}) + 34 u^{2} (2 u) + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.5.2.2

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

$256 u^{6} + 32 \cdot 2 u^{1 + 4} + 32 u^{4} \cdot - 3 - 32 u^{3} (8 u^{2}) - 32 u^{3} (2 u) - 32 u^{3} \cdot - 3 + 34 u^{2} (8 u^{2}) + 34 u^{2} (2 u) + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.5.3

Add $1$ and $4$ .

$256 u^{6} + 32 \cdot 2 u^{5} + 32 u^{4} \cdot - 3 - 32 u^{3} (8 u^{2}) - 32 u^{3} (2 u) - 32 u^{3} \cdot - 3 + 34 u^{2} (8 u^{2}) + 34 u^{2} (2 u) + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.6

Multiply $32$ by $2$ .

$256 u^{6} + 64 u^{5} + 32 u^{4} \cdot - 3 - 32 u^{3} (8 u^{2}) - 32 u^{3} (2 u) - 32 u^{3} \cdot - 3 + 34 u^{2} (8 u^{2}) + 34 u^{2} (2 u) + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.7

Multiply $- 3$ by $32$ .

$256 u^{6} + 64 u^{5} - 96 u^{4} - 32 u^{3} (8 u^{2}) - 32 u^{3} (2 u) - 32 u^{3} \cdot - 3 + 34 u^{2} (8 u^{2}) + 34 u^{2} (2 u) + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.8

Rewrite using the commutative property of multiplication.

$256 u^{6} + 64 u^{5} - 96 u^{4} - 32 \cdot 8 u^{3} u^{2} - 32 u^{3} (2 u) - 32 u^{3} \cdot - 3 + 34 u^{2} (8 u^{2}) + 34 u^{2} (2 u) + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.9

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

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Step 4.1.9.1

Move $u^{2}$ .

$256 u^{6} + 64 u^{5} - 96 u^{4} - 32 \cdot 8 (u^{2} u^{3}) - 32 u^{3} (2 u) - 32 u^{3} \cdot - 3 + 34 u^{2} (8 u^{2}) + 34 u^{2} (2 u) + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.9.2

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

$256 u^{6} + 64 u^{5} - 96 u^{4} - 32 \cdot 8 u^{2 + 3} - 32 u^{3} (2 u) - 32 u^{3} \cdot - 3 + 34 u^{2} (8 u^{2}) + 34 u^{2} (2 u) + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.9.3

Add $2$ and $3$ .

$256 u^{6} + 64 u^{5} - 96 u^{4} - 32 \cdot 8 u^{5} - 32 u^{3} (2 u) - 32 u^{3} \cdot - 3 + 34 u^{2} (8 u^{2}) + 34 u^{2} (2 u) + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.10

Multiply $- 32$ by $8$ .

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 32 u^{3} (2 u) - 32 u^{3} \cdot - 3 + 34 u^{2} (8 u^{2}) + 34 u^{2} (2 u) + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.11

Rewrite using the commutative property of multiplication.

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 32 \cdot 2 u^{3} u - 32 u^{3} \cdot - 3 + 34 u^{2} (8 u^{2}) + 34 u^{2} (2 u) + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.12

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

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Step 4.1.12.1

Move $u$ .

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 32 \cdot 2 (u \cdot u^{3}) - 32 u^{3} \cdot - 3 + 34 u^{2} (8 u^{2}) + 34 u^{2} (2 u) + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.12.2

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

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Step 4.1.12.2.1

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

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 32 \cdot 2 (u^{1} u^{3}) - 32 u^{3} \cdot - 3 + 34 u^{2} (8 u^{2}) + 34 u^{2} (2 u) + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.12.2.2

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

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 32 \cdot 2 u^{1 + 3} - 32 u^{3} \cdot - 3 + 34 u^{2} (8 u^{2}) + 34 u^{2} (2 u) + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.12.3

Add $1$ and $3$ .

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 32 \cdot 2 u^{4} - 32 u^{3} \cdot - 3 + 34 u^{2} (8 u^{2}) + 34 u^{2} (2 u) + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.13

Multiply $- 32$ by $2$ .

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 64 u^{4} - 32 u^{3} \cdot - 3 + 34 u^{2} (8 u^{2}) + 34 u^{2} (2 u) + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.14

Multiply $- 3$ by $- 32$ .

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 64 u^{4} + 96 u^{3} + 34 u^{2} (8 u^{2}) + 34 u^{2} (2 u) + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.15

Rewrite using the commutative property of multiplication.

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 64 u^{4} + 96 u^{3} + 34 \cdot 8 u^{2} u^{2} + 34 u^{2} (2 u) + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.16

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

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Step 4.1.16.1

Move $u^{2}$ .

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 64 u^{4} + 96 u^{3} + 34 \cdot 8 (u^{2} u^{2}) + 34 u^{2} (2 u) + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.16.2

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

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 64 u^{4} + 96 u^{3} + 34 \cdot 8 u^{2 + 2} + 34 u^{2} (2 u) + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.16.3

Add $2$ and $2$ .

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 64 u^{4} + 96 u^{3} + 34 \cdot 8 u^{4} + 34 u^{2} (2 u) + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.17

Multiply $34$ by $8$ .

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 64 u^{4} + 96 u^{3} + 272 u^{4} + 34 u^{2} (2 u) + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.18

Rewrite using the commutative property of multiplication.

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 64 u^{4} + 96 u^{3} + 272 u^{4} + 34 \cdot 2 u^{2} u + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.19

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

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Step 4.1.19.1

Move $u$ .

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 64 u^{4} + 96 u^{3} + 272 u^{4} + 34 \cdot 2 (u \cdot u^{2}) + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.19.2

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

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Step 4.1.19.2.1

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

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 64 u^{4} + 96 u^{3} + 272 u^{4} + 34 \cdot 2 (u^{1} u^{2}) + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.19.2.2

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

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 64 u^{4} + 96 u^{3} + 272 u^{4} + 34 \cdot 2 u^{1 + 2} + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.19.3

Add $1$ and $2$ .

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 64 u^{4} + 96 u^{3} + 272 u^{4} + 34 \cdot 2 u^{3} + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.20

Multiply $34$ by $2$ .

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 64 u^{4} + 96 u^{3} + 272 u^{4} + 68 u^{3} + 34 u^{2} \cdot - 3 + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.21

Multiply $- 3$ by $34$ .

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 64 u^{4} + 96 u^{3} + 272 u^{4} + 68 u^{3} - 102 u^{2} + 27 u (8 u^{2}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.22

Rewrite using the commutative property of multiplication.

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 64 u^{4} + 96 u^{3} + 272 u^{4} + 68 u^{3} - 102 u^{2} + 27 \cdot 8 u \cdot u^{2} + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.23

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

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Step 4.1.23.1

Move $u^{2}$ .

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 64 u^{4} + 96 u^{3} + 272 u^{4} + 68 u^{3} - 102 u^{2} + 27 \cdot 8 (u^{2} u) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.23.2

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

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Step 4.1.23.2.1

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

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 64 u^{4} + 96 u^{3} + 272 u^{4} + 68 u^{3} - 102 u^{2} + 27 \cdot 8 (u^{2} u^{1}) + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.23.2.2

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

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 64 u^{4} + 96 u^{3} + 272 u^{4} + 68 u^{3} - 102 u^{2} + 27 \cdot 8 u^{2 + 1} + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.23.3

Add $2$ and $1$ .

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 64 u^{4} + 96 u^{3} + 272 u^{4} + 68 u^{3} - 102 u^{2} + 27 \cdot 8 u^{3} + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.24

Multiply $27$ by $8$ .

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 64 u^{4} + 96 u^{3} + 272 u^{4} + 68 u^{3} - 102 u^{2} + 216 u^{3} + 27 u (2 u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.25

Rewrite using the commutative property of multiplication.

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 64 u^{4} + 96 u^{3} + 272 u^{4} + 68 u^{3} - 102 u^{2} + 216 u^{3} + 27 \cdot 2 u \cdot u + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.26

Multiply $u$ by $u$ by adding the exponents.

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Step 4.1.26.1

Move $u$ .

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 64 u^{4} + 96 u^{3} + 272 u^{4} + 68 u^{3} - 102 u^{2} + 216 u^{3} + 27 \cdot 2 (u \cdot u) + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.26.2

Multiply $u$ by $u$ .

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 64 u^{4} + 96 u^{3} + 272 u^{4} + 68 u^{3} - 102 u^{2} + 216 u^{3} + 27 \cdot 2 u^{2} + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.27

Multiply $27$ by $2$ .

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 64 u^{4} + 96 u^{3} + 272 u^{4} + 68 u^{3} - 102 u^{2} + 216 u^{3} + 54 u^{2} + 27 u \cdot - 3 + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.28

Multiply $- 3$ by $27$ .

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 64 u^{4} + 96 u^{3} + 272 u^{4} + 68 u^{3} - 102 u^{2} + 216 u^{3} + 54 u^{2} - 81 u + 24 (8 u^{2}) + 24 (2 u) + 24 \cdot - 3$

Step 4.1.29

Multiply $8$ by $24$ .

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 64 u^{4} + 96 u^{3} + 272 u^{4} + 68 u^{3} - 102 u^{2} + 216 u^{3} + 54 u^{2} - 81 u + 192 u^{2} + 24 (2 u) + 24 \cdot - 3$

Step 4.1.30

Multiply $2$ by $24$ .

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 64 u^{4} + 96 u^{3} + 272 u^{4} + 68 u^{3} - 102 u^{2} + 216 u^{3} + 54 u^{2} - 81 u + 192 u^{2} + 48 u + 24 \cdot - 3$

Step 4.1.31

Multiply $24$ by $- 3$ .

$256 u^{6} + 64 u^{5} - 96 u^{4} - 256 u^{5} - 64 u^{4} + 96 u^{3} + 272 u^{4} + 68 u^{3} - 102 u^{2} + 216 u^{3} + 54 u^{2} - 81 u + 192 u^{2} + 48 u - 72$

Step 4.2

Simplify by adding terms.

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Step 4.2.1

Subtract $256 u^{5}$ from $64 u^{5}$ .

$256 u^{6} - 192 u^{5} - 96 u^{4} - 64 u^{4} + 96 u^{3} + 272 u^{4} + 68 u^{3} - 102 u^{2} + 216 u^{3} + 54 u^{2} - 81 u + 192 u^{2} + 48 u - 72$

Step 4.2.2

Subtract $64 u^{4}$ from $- 96 u^{4}$ .

$256 u^{6} - 192 u^{5} - 160 u^{4} + 96 u^{3} + 272 u^{4} + 68 u^{3} - 102 u^{2} + 216 u^{3} + 54 u^{2} - 81 u + 192 u^{2} + 48 u - 72$

Step 4.2.3

Add $- 160 u^{4}$ and $272 u^{4}$ .

$256 u^{6} - 192 u^{5} + 112 u^{4} + 96 u^{3} + 68 u^{3} - 102 u^{2} + 216 u^{3} + 54 u^{2} - 81 u + 192 u^{2} + 48 u - 72$

Step 4.2.4

Add $96 u^{3}$ and $68 u^{3}$ .

$256 u^{6} - 192 u^{5} + 112 u^{4} + 164 u^{3} - 102 u^{2} + 216 u^{3} + 54 u^{2} - 81 u + 192 u^{2} + 48 u - 72$

Step 4.2.5

Add $164 u^{3}$ and $216 u^{3}$ .

$256 u^{6} - 192 u^{5} + 112 u^{4} + 380 u^{3} - 102 u^{2} + 54 u^{2} - 81 u + 192 u^{2} + 48 u - 72$

Step 4.2.6

Add $- 102 u^{2}$ and $54 u^{2}$ .

$256 u^{6} - 192 u^{5} + 112 u^{4} + 380 u^{3} - 48 u^{2} - 81 u + 192 u^{2} + 48 u - 72$

Step 4.2.7

Add $- 48 u^{2}$ and $192 u^{2}$ .

$256 u^{6} - 192 u^{5} + 112 u^{4} + 380 u^{3} + 144 u^{2} - 81 u + 48 u - 72$

Step 4.2.8

Add $- 81 u$ and $48 u$ .

$256 u^{6} - 192 u^{5} + 112 u^{4} + 380 u^{3} + 144 u^{2} - 33 u - 72$