Algebra Examples

${(3 a + 4 b)}^{8}$

Step 1

Pascal's Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$\dots$

$1 - 6 - 15 - 20 - 15 - 6 - 1$

$1 - 7 - 21 - 35 - 35 - 21 - 7 - 1$

$1 - 8 - 28 - 56 - 70 - 56 - 28 - 8 - 1$

The triangle can be used to calculate the coefficients of the expansion of ${(a + b)}^{n}$ by taking the exponent $n$ and adding $1$ . The coefficients will correspond with line $n + 1$ of the triangle. For ${(3 a + 4 b)}^{8}$ , $n = 8$ so the coefficients of the expansion will correspond with line $9$ .

Step 2

The expansion follows the rule ${(a + b)}^{n} = c_{0} a^{n} b^{0} + c_{1} a^{n - 1} b^{1} + c_{n - 1} a^{1} b^{n - 1} + c_{n} a^{0} b^{n}$ . The values of the coefficients, from the triangle, are $1 - 8 - 28 - 56 - 70 - 56 - 28 - 8 - 1$ .

$1 a^{8} b^{0} + 8 a^{7} b + 28 a^{6} b^{2} + 56 a^{5} b^{3} + 70 a^{4} b^{4} + 56 a^{3} b^{5} + 28 a^{2} b^{6} + 8 a b^{7} + 1 a^{0} b^{8}$

Step 3

Substitute the actual values of $a$ $3 a$ and $b$ $4 b$ into the expression.

$1 {(3 a)}^{8} {(4 b)}^{0} + 8 {(3 a)}^{7} {(4 b)}^{1} + 28 {(3 a)}^{6} {(4 b)}^{2} + 56 {(3 a)}^{5} {(4 b)}^{3} + 70 {(3 a)}^{4} {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4

Simplify each term.

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Step 4.1

Multiply ${(3 a)}^{8}$ by $1$ .

${(3 a)}^{8} {(4 b)}^{0} + 8 {(3 a)}^{7} {(4 b)}^{1} + 28 {(3 a)}^{6} {(4 b)}^{2} + 56 {(3 a)}^{5} {(4 b)}^{3} + 70 {(3 a)}^{4} {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.2

Apply the product rule to $3 a$ .

$3^{8} a^{8} {(4 b)}^{0} + 8 {(3 a)}^{7} {(4 b)}^{1} + 28 {(3 a)}^{6} {(4 b)}^{2} + 56 {(3 a)}^{5} {(4 b)}^{3} + 70 {(3 a)}^{4} {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.3

Raise $3$ to the power of $8$ .

$6561 a^{8} {(4 b)}^{0} + 8 {(3 a)}^{7} {(4 b)}^{1} + 28 {(3 a)}^{6} {(4 b)}^{2} + 56 {(3 a)}^{5} {(4 b)}^{3} + 70 {(3 a)}^{4} {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.4

Apply the product rule to $4 b$ .

$6561 a^{8} (4^{0} b^{0}) + 8 {(3 a)}^{7} {(4 b)}^{1} + 28 {(3 a)}^{6} {(4 b)}^{2} + 56 {(3 a)}^{5} {(4 b)}^{3} + 70 {(3 a)}^{4} {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.5

Rewrite using the commutative property of multiplication.

$6561 \cdot 4^{0} a^{8} b^{0} + 8 {(3 a)}^{7} {(4 b)}^{1} + 28 {(3 a)}^{6} {(4 b)}^{2} + 56 {(3 a)}^{5} {(4 b)}^{3} + 70 {(3 a)}^{4} {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.6

Anything raised to $0$ is $1$ .

$6561 \cdot 1 a^{8} b^{0} + 8 {(3 a)}^{7} {(4 b)}^{1} + 28 {(3 a)}^{6} {(4 b)}^{2} + 56 {(3 a)}^{5} {(4 b)}^{3} + 70 {(3 a)}^{4} {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.7

Multiply $6561$ by $1$ .

$6561 a^{8} b^{0} + 8 {(3 a)}^{7} {(4 b)}^{1} + 28 {(3 a)}^{6} {(4 b)}^{2} + 56 {(3 a)}^{5} {(4 b)}^{3} + 70 {(3 a)}^{4} {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.8

Anything raised to $0$ is $1$ .

$6561 a^{8} \cdot 1 + 8 {(3 a)}^{7} {(4 b)}^{1} + 28 {(3 a)}^{6} {(4 b)}^{2} + 56 {(3 a)}^{5} {(4 b)}^{3} + 70 {(3 a)}^{4} {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.9

Multiply $6561$ by $1$ .

$6561 a^{8} + 8 {(3 a)}^{7} {(4 b)}^{1} + 28 {(3 a)}^{6} {(4 b)}^{2} + 56 {(3 a)}^{5} {(4 b)}^{3} + 70 {(3 a)}^{4} {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.10

Apply the product rule to $3 a$ .

$6561 a^{8} + 8 (3^{7} a^{7}) {(4 b)}^{1} + 28 {(3 a)}^{6} {(4 b)}^{2} + 56 {(3 a)}^{5} {(4 b)}^{3} + 70 {(3 a)}^{4} {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.11

Raise $3$ to the power of $7$ .

$6561 a^{8} + 8 (2187 a^{7}) {(4 b)}^{1} + 28 {(3 a)}^{6} {(4 b)}^{2} + 56 {(3 a)}^{5} {(4 b)}^{3} + 70 {(3 a)}^{4} {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.12

Multiply $2187$ by $8$ .

$6561 a^{8} + 17496 a^{7} {(4 b)}^{1} + 28 {(3 a)}^{6} {(4 b)}^{2} + 56 {(3 a)}^{5} {(4 b)}^{3} + 70 {(3 a)}^{4} {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.13

Simplify.

$6561 a^{8} + 17496 a^{7} (4 b) + 28 {(3 a)}^{6} {(4 b)}^{2} + 56 {(3 a)}^{5} {(4 b)}^{3} + 70 {(3 a)}^{4} {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.14

Rewrite using the commutative property of multiplication.

$6561 a^{8} + 17496 \cdot 4 a^{7} b + 28 {(3 a)}^{6} {(4 b)}^{2} + 56 {(3 a)}^{5} {(4 b)}^{3} + 70 {(3 a)}^{4} {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.15

Multiply $17496$ by $4$ .

$6561 a^{8} + 69984 a^{7} b + 28 {(3 a)}^{6} {(4 b)}^{2} + 56 {(3 a)}^{5} {(4 b)}^{3} + 70 {(3 a)}^{4} {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.16

Apply the product rule to $3 a$ .

$6561 a^{8} + 69984 a^{7} b + 28 (3^{6} a^{6}) {(4 b)}^{2} + 56 {(3 a)}^{5} {(4 b)}^{3} + 70 {(3 a)}^{4} {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.17

Raise $3$ to the power of $6$ .

$6561 a^{8} + 69984 a^{7} b + 28 (729 a^{6}) {(4 b)}^{2} + 56 {(3 a)}^{5} {(4 b)}^{3} + 70 {(3 a)}^{4} {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.18

Multiply $729$ by $28$ .

$6561 a^{8} + 69984 a^{7} b + 20412 a^{6} {(4 b)}^{2} + 56 {(3 a)}^{5} {(4 b)}^{3} + 70 {(3 a)}^{4} {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.19

Apply the product rule to $4 b$ .

$6561 a^{8} + 69984 a^{7} b + 20412 a^{6} (4^{2} b^{2}) + 56 {(3 a)}^{5} {(4 b)}^{3} + 70 {(3 a)}^{4} {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.20

Rewrite using the commutative property of multiplication.

$6561 a^{8} + 69984 a^{7} b + 20412 \cdot 4^{2} a^{6} b^{2} + 56 {(3 a)}^{5} {(4 b)}^{3} + 70 {(3 a)}^{4} {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.21

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

$6561 a^{8} + 69984 a^{7} b + 20412 \cdot 16 a^{6} b^{2} + 56 {(3 a)}^{5} {(4 b)}^{3} + 70 {(3 a)}^{4} {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.22

Multiply $20412$ by $16$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 56 {(3 a)}^{5} {(4 b)}^{3} + 70 {(3 a)}^{4} {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.23

Apply the product rule to $3 a$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 56 (3^{5} a^{5}) {(4 b)}^{3} + 70 {(3 a)}^{4} {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.24

Raise $3$ to the power of $5$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 56 (243 a^{5}) {(4 b)}^{3} + 70 {(3 a)}^{4} {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.25

Multiply $243$ by $56$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 13608 a^{5} {(4 b)}^{3} + 70 {(3 a)}^{4} {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.26

Apply the product rule to $4 b$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 13608 a^{5} (4^{3} b^{3}) + 70 {(3 a)}^{4} {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.27

Rewrite using the commutative property of multiplication.

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 13608 \cdot 4^{3} a^{5} b^{3} + 70 {(3 a)}^{4} {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.28

Raise $4$ to the power of $3$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 13608 \cdot 64 a^{5} b^{3} + 70 {(3 a)}^{4} {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.29

Multiply $13608$ by $64$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 70 {(3 a)}^{4} {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.30

Apply the product rule to $3 a$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 70 (3^{4} a^{4}) {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.31

Raise $3$ to the power of $4$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 70 (81 a^{4}) {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.32

Multiply $81$ by $70$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 5670 a^{4} {(4 b)}^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.33

Apply the product rule to $4 b$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 5670 a^{4} (4^{4} b^{4}) + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.34

Rewrite using the commutative property of multiplication.

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 5670 \cdot 4^{4} a^{4} b^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.35

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

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 5670 \cdot 256 a^{4} b^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.36

Multiply $5670$ by $256$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 1451520 a^{4} b^{4} + 56 {(3 a)}^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.37

Apply the product rule to $3 a$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 1451520 a^{4} b^{4} + 56 (3^{3} a^{3}) {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.38

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

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 1451520 a^{4} b^{4} + 56 (27 a^{3}) {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.39

Multiply $27$ by $56$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 1451520 a^{4} b^{4} + 1512 a^{3} {(4 b)}^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.40

Apply the product rule to $4 b$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 1451520 a^{4} b^{4} + 1512 a^{3} (4^{5} b^{5}) + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.41

Rewrite using the commutative property of multiplication.

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 1451520 a^{4} b^{4} + 1512 \cdot 4^{5} a^{3} b^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.42

Raise $4$ to the power of $5$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 1451520 a^{4} b^{4} + 1512 \cdot 1024 a^{3} b^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.43

Multiply $1512$ by $1024$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 1451520 a^{4} b^{4} + 1548288 a^{3} b^{5} + 28 {(3 a)}^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.44

Apply the product rule to $3 a$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 1451520 a^{4} b^{4} + 1548288 a^{3} b^{5} + 28 (3^{2} a^{2}) {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.45

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

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 1451520 a^{4} b^{4} + 1548288 a^{3} b^{5} + 28 (9 a^{2}) {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.46

Multiply $9$ by $28$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 1451520 a^{4} b^{4} + 1548288 a^{3} b^{5} + 252 a^{2} {(4 b)}^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.47

Apply the product rule to $4 b$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 1451520 a^{4} b^{4} + 1548288 a^{3} b^{5} + 252 a^{2} (4^{6} b^{6}) + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.48

Rewrite using the commutative property of multiplication.

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 1451520 a^{4} b^{4} + 1548288 a^{3} b^{5} + 252 \cdot 4^{6} a^{2} b^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.49

Raise $4$ to the power of $6$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 1451520 a^{4} b^{4} + 1548288 a^{3} b^{5} + 252 \cdot 4096 a^{2} b^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.50

Multiply $252$ by $4096$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 1451520 a^{4} b^{4} + 1548288 a^{3} b^{5} + 1032192 a^{2} b^{6} + 8 {(3 a)}^{1} {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.51

Simplify.

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 1451520 a^{4} b^{4} + 1548288 a^{3} b^{5} + 1032192 a^{2} b^{6} + 8 (3 a) {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.52

Multiply $3$ by $8$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 1451520 a^{4} b^{4} + 1548288 a^{3} b^{5} + 1032192 a^{2} b^{6} + 24 a {(4 b)}^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.53

Apply the product rule to $4 b$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 1451520 a^{4} b^{4} + 1548288 a^{3} b^{5} + 1032192 a^{2} b^{6} + 24 a (4^{7} b^{7}) + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.54

Rewrite using the commutative property of multiplication.

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 1451520 a^{4} b^{4} + 1548288 a^{3} b^{5} + 1032192 a^{2} b^{6} + 24 \cdot 4^{7} a b^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.55

Raise $4$ to the power of $7$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 1451520 a^{4} b^{4} + 1548288 a^{3} b^{5} + 1032192 a^{2} b^{6} + 24 \cdot 16384 a b^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.56

Multiply $24$ by $16384$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 1451520 a^{4} b^{4} + 1548288 a^{3} b^{5} + 1032192 a^{2} b^{6} + 393216 a b^{7} + 1 {(3 a)}^{0} {(4 b)}^{8}$

Step 4.57

Multiply ${(3 a)}^{0}$ by $1$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 1451520 a^{4} b^{4} + 1548288 a^{3} b^{5} + 1032192 a^{2} b^{6} + 393216 a b^{7} + {(3 a)}^{0} {(4 b)}^{8}$

Step 4.58

Apply the product rule to $3 a$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 1451520 a^{4} b^{4} + 1548288 a^{3} b^{5} + 1032192 a^{2} b^{6} + 393216 a b^{7} + 3^{0} a^{0} {(4 b)}^{8}$

Step 4.59

Anything raised to $0$ is $1$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 1451520 a^{4} b^{4} + 1548288 a^{3} b^{5} + 1032192 a^{2} b^{6} + 393216 a b^{7} + 1 a^{0} {(4 b)}^{8}$

Step 4.60

Multiply $a^{0}$ by $1$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 1451520 a^{4} b^{4} + 1548288 a^{3} b^{5} + 1032192 a^{2} b^{6} + 393216 a b^{7} + a^{0} {(4 b)}^{8}$

Step 4.61

Anything raised to $0$ is $1$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 1451520 a^{4} b^{4} + 1548288 a^{3} b^{5} + 1032192 a^{2} b^{6} + 393216 a b^{7} + 1 {(4 b)}^{8}$

Step 4.62

Multiply ${(4 b)}^{8}$ by $1$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 1451520 a^{4} b^{4} + 1548288 a^{3} b^{5} + 1032192 a^{2} b^{6} + 393216 a b^{7} + {(4 b)}^{8}$

Step 4.63

Apply the product rule to $4 b$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 1451520 a^{4} b^{4} + 1548288 a^{3} b^{5} + 1032192 a^{2} b^{6} + 393216 a b^{7} + 4^{8} b^{8}$

Step 4.64

Raise $4$ to the power of $8$ .

$6561 a^{8} + 69984 a^{7} b + 326592 a^{6} b^{2} + 870912 a^{5} b^{3} + 1451520 a^{4} b^{4} + 1548288 a^{3} b^{5} + 1032192 a^{2} b^{6} + 393216 a b^{7} + 65536 b^{8}$