Precalculus Examples

${(2 x - 3 y)}^{11}$

Step 1

Pascal's Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$\dots$

$1 - 9 - 36 - 84 - 126 - 126 - 84 - 36 - 9 - 1$

$1 - 10 - 45 - 120 - 210 - 252 - 210 - 120 - 45 - 10 - 1$

$1 - 11 - 55 - 165 - 330 - 462 - 462 - 330 - 165 - 55 - 11 - 1$

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 ${(2 x - 3 y)}^{11}$ , $n = 11$ so the coefficients of the expansion will correspond with line $12$ .

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 $1 - 11 - 55 - 165 - 330 - 462 - 462 - 330 - 165 - 55 - 11 - 1$ .

$1 a^{11} b^{0} + 11 a^{10} b + 55 a^{9} b^{2} + 165 a^{8} b^{3} + 330 a^{7} b^{4} + 462 a^{6} b^{5} + 462 a^{5} b^{6} + 330 a^{4} b^{7} + 165 a^{3} b^{8} + 55 a^{2} b^{9} + 11 a b^{10} + 1 a^{0} b^{11}$

Step 3

Substitute the actual values of $a$ $2 x$ and $b$ $- 3 y$ into the expression.

$1 {(2 x)}^{11} {(- 3 y)}^{0} + 11 {(2 x)}^{10} {(- 3 y)}^{1} + 55 {(2 x)}^{9} {(- 3 y)}^{2} + 165 {(2 x)}^{8} {(- 3 y)}^{3} + 330 {(2 x)}^{7} {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4

Simplify each term.

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Step 4.1

Multiply ${(2 x)}^{11}$ by $1$ .

${(2 x)}^{11} {(- 3 y)}^{0} + 11 {(2 x)}^{10} {(- 3 y)}^{1} + 55 {(2 x)}^{9} {(- 3 y)}^{2} + 165 {(2 x)}^{8} {(- 3 y)}^{3} + 330 {(2 x)}^{7} {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.2

Apply the product rule to $2 x$ .

$2^{11} x^{11} {(- 3 y)}^{0} + 11 {(2 x)}^{10} {(- 3 y)}^{1} + 55 {(2 x)}^{9} {(- 3 y)}^{2} + 165 {(2 x)}^{8} {(- 3 y)}^{3} + 330 {(2 x)}^{7} {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.3

Raise $2$ to the power of $11$ .

$2048 x^{11} {(- 3 y)}^{0} + 11 {(2 x)}^{10} {(- 3 y)}^{1} + 55 {(2 x)}^{9} {(- 3 y)}^{2} + 165 {(2 x)}^{8} {(- 3 y)}^{3} + 330 {(2 x)}^{7} {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.4

Apply the product rule to $- 3 y$ .

$2048 x^{11} ({(- 3)}^{0} y^{0}) + 11 {(2 x)}^{10} {(- 3 y)}^{1} + 55 {(2 x)}^{9} {(- 3 y)}^{2} + 165 {(2 x)}^{8} {(- 3 y)}^{3} + 330 {(2 x)}^{7} {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.5

Rewrite using the commutative property of multiplication.

$2048 \cdot {(- 3)}^{0} x^{11} y^{0} + 11 {(2 x)}^{10} {(- 3 y)}^{1} + 55 {(2 x)}^{9} {(- 3 y)}^{2} + 165 {(2 x)}^{8} {(- 3 y)}^{3} + 330 {(2 x)}^{7} {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.6

Anything raised to $0$ is $1$ .

$2048 \cdot 1 x^{11} y^{0} + 11 {(2 x)}^{10} {(- 3 y)}^{1} + 55 {(2 x)}^{9} {(- 3 y)}^{2} + 165 {(2 x)}^{8} {(- 3 y)}^{3} + 330 {(2 x)}^{7} {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.7

Multiply $2048$ by $1$ .

$2048 x^{11} y^{0} + 11 {(2 x)}^{10} {(- 3 y)}^{1} + 55 {(2 x)}^{9} {(- 3 y)}^{2} + 165 {(2 x)}^{8} {(- 3 y)}^{3} + 330 {(2 x)}^{7} {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.8

Anything raised to $0$ is $1$ .

$2048 x^{11} \cdot 1 + 11 {(2 x)}^{10} {(- 3 y)}^{1} + 55 {(2 x)}^{9} {(- 3 y)}^{2} + 165 {(2 x)}^{8} {(- 3 y)}^{3} + 330 {(2 x)}^{7} {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.9

Multiply $2048$ by $1$ .

$2048 x^{11} + 11 {(2 x)}^{10} {(- 3 y)}^{1} + 55 {(2 x)}^{9} {(- 3 y)}^{2} + 165 {(2 x)}^{8} {(- 3 y)}^{3} + 330 {(2 x)}^{7} {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.10

Apply the product rule to $2 x$ .

$2048 x^{11} + 11 (2^{10} x^{10}) {(- 3 y)}^{1} + 55 {(2 x)}^{9} {(- 3 y)}^{2} + 165 {(2 x)}^{8} {(- 3 y)}^{3} + 330 {(2 x)}^{7} {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.11

Raise $2$ to the power of $10$ .

$2048 x^{11} + 11 (1024 x^{10}) {(- 3 y)}^{1} + 55 {(2 x)}^{9} {(- 3 y)}^{2} + 165 {(2 x)}^{8} {(- 3 y)}^{3} + 330 {(2 x)}^{7} {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.12

Multiply $1024$ by $11$ .

$2048 x^{11} + 11264 x^{10} {(- 3 y)}^{1} + 55 {(2 x)}^{9} {(- 3 y)}^{2} + 165 {(2 x)}^{8} {(- 3 y)}^{3} + 330 {(2 x)}^{7} {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.13

Simplify.

$2048 x^{11} + 11264 x^{10} (- 3 y) + 55 {(2 x)}^{9} {(- 3 y)}^{2} + 165 {(2 x)}^{8} {(- 3 y)}^{3} + 330 {(2 x)}^{7} {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.14

Rewrite using the commutative property of multiplication.

$2048 x^{11} + 11264 \cdot - 3 x^{10} y + 55 {(2 x)}^{9} {(- 3 y)}^{2} + 165 {(2 x)}^{8} {(- 3 y)}^{3} + 330 {(2 x)}^{7} {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.15

Multiply $11264$ by $- 3$ .

$2048 x^{11} - 33792 x^{10} y + 55 {(2 x)}^{9} {(- 3 y)}^{2} + 165 {(2 x)}^{8} {(- 3 y)}^{3} + 330 {(2 x)}^{7} {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.16

Apply the product rule to $2 x$ .

$2048 x^{11} - 33792 x^{10} y + 55 (2^{9} x^{9}) {(- 3 y)}^{2} + 165 {(2 x)}^{8} {(- 3 y)}^{3} + 330 {(2 x)}^{7} {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.17

Raise $2$ to the power of $9$ .

$2048 x^{11} - 33792 x^{10} y + 55 (512 x^{9}) {(- 3 y)}^{2} + 165 {(2 x)}^{8} {(- 3 y)}^{3} + 330 {(2 x)}^{7} {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.18

Multiply $512$ by $55$ .

$2048 x^{11} - 33792 x^{10} y + 28160 x^{9} {(- 3 y)}^{2} + 165 {(2 x)}^{8} {(- 3 y)}^{3} + 330 {(2 x)}^{7} {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.19

Apply the product rule to $- 3 y$ .

$2048 x^{11} - 33792 x^{10} y + 28160 x^{9} ({(- 3)}^{2} y^{2}) + 165 {(2 x)}^{8} {(- 3 y)}^{3} + 330 {(2 x)}^{7} {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.20

Rewrite using the commutative property of multiplication.

$2048 x^{11} - 33792 x^{10} y + 28160 \cdot {(- 3)}^{2} x^{9} y^{2} + 165 {(2 x)}^{8} {(- 3 y)}^{3} + 330 {(2 x)}^{7} {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.21

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

$2048 x^{11} - 33792 x^{10} y + 28160 \cdot 9 x^{9} y^{2} + 165 {(2 x)}^{8} {(- 3 y)}^{3} + 330 {(2 x)}^{7} {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.22

Multiply $28160$ by $9$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} + 165 {(2 x)}^{8} {(- 3 y)}^{3} + 330 {(2 x)}^{7} {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.23

Apply the product rule to $2 x$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} + 165 (2^{8} x^{8}) {(- 3 y)}^{3} + 330 {(2 x)}^{7} {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.24

Raise $2$ to the power of $8$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} + 165 (256 x^{8}) {(- 3 y)}^{3} + 330 {(2 x)}^{7} {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.25

Multiply $256$ by $165$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} + 42240 x^{8} {(- 3 y)}^{3} + 330 {(2 x)}^{7} {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.26

Apply the product rule to $- 3 y$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} + 42240 x^{8} ({(- 3)}^{3} y^{3}) + 330 {(2 x)}^{7} {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.27

Rewrite using the commutative property of multiplication.

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} + 42240 \cdot {(- 3)}^{3} x^{8} y^{3} + 330 {(2 x)}^{7} {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.28

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

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} + 42240 \cdot - 27 x^{8} y^{3} + 330 {(2 x)}^{7} {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.29

Multiply $42240$ by $- 27$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 330 {(2 x)}^{7} {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.30

Apply the product rule to $2 x$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 330 (2^{7} x^{7}) {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.31

Raise $2$ to the power of $7$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 330 (128 x^{7}) {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.32

Multiply $128$ by $330$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 42240 x^{7} {(- 3 y)}^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.33

Apply the product rule to $- 3 y$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 42240 x^{7} ({(- 3)}^{4} y^{4}) + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.34

Rewrite using the commutative property of multiplication.

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 42240 \cdot {(- 3)}^{4} x^{7} y^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.35

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

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 42240 \cdot 81 x^{7} y^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.36

Multiply $42240$ by $81$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} + 462 {(2 x)}^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.37

Apply the product rule to $2 x$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} + 462 (2^{6} x^{6}) {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.38

Raise $2$ to the power of $6$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} + 462 (64 x^{6}) {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.39

Multiply $64$ by $462$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} + 29568 x^{6} {(- 3 y)}^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.40

Apply the product rule to $- 3 y$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} + 29568 x^{6} ({(- 3)}^{5} y^{5}) + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.41

Rewrite using the commutative property of multiplication.

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} + 29568 \cdot {(- 3)}^{5} x^{6} y^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.42

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

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} + 29568 \cdot - 243 x^{6} y^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.43

Multiply $29568$ by $- 243$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 462 {(2 x)}^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.44

Apply the product rule to $2 x$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 462 (2^{5} x^{5}) {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.45

Raise $2$ to the power of $5$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 462 (32 x^{5}) {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.46

Multiply $32$ by $462$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 14784 x^{5} {(- 3 y)}^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.47

Apply the product rule to $- 3 y$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 14784 x^{5} ({(- 3)}^{6} y^{6}) + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.48

Rewrite using the commutative property of multiplication.

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 14784 \cdot {(- 3)}^{6} x^{5} y^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.49

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

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 14784 \cdot 729 x^{5} y^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.50

Multiply $14784$ by $729$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} + 330 {(2 x)}^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.51

Apply the product rule to $2 x$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} + 330 (2^{4} x^{4}) {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.52

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

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} + 330 (16 x^{4}) {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.53

Multiply $16$ by $330$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} + 5280 x^{4} {(- 3 y)}^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.54

Apply the product rule to $- 3 y$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} + 5280 x^{4} ({(- 3)}^{7} y^{7}) + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.55

Rewrite using the commutative property of multiplication.

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} + 5280 \cdot {(- 3)}^{7} x^{4} y^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.56

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

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} + 5280 \cdot - 2187 x^{4} y^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.57

Multiply $5280$ by $- 2187$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} - 11547360 x^{4} y^{7} + 165 {(2 x)}^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.58

Apply the product rule to $2 x$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} - 11547360 x^{4} y^{7} + 165 (2^{3} x^{3}) {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.59

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

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} - 11547360 x^{4} y^{7} + 165 (8 x^{3}) {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.60

Multiply $8$ by $165$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} - 11547360 x^{4} y^{7} + 1320 x^{3} {(- 3 y)}^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.61

Apply the product rule to $- 3 y$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} - 11547360 x^{4} y^{7} + 1320 x^{3} ({(- 3)}^{8} y^{8}) + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.62

Rewrite using the commutative property of multiplication.

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} - 11547360 x^{4} y^{7} + 1320 \cdot {(- 3)}^{8} x^{3} y^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.63

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

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} - 11547360 x^{4} y^{7} + 1320 \cdot 6561 x^{3} y^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.64

Multiply $1320$ by $6561$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} - 11547360 x^{4} y^{7} + 8660520 x^{3} y^{8} + 55 {(2 x)}^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.65

Apply the product rule to $2 x$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} - 11547360 x^{4} y^{7} + 8660520 x^{3} y^{8} + 55 (2^{2} x^{2}) {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.66

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

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} - 11547360 x^{4} y^{7} + 8660520 x^{3} y^{8} + 55 (4 x^{2}) {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.67

Multiply $4$ by $55$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} - 11547360 x^{4} y^{7} + 8660520 x^{3} y^{8} + 220 x^{2} {(- 3 y)}^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.68

Apply the product rule to $- 3 y$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} - 11547360 x^{4} y^{7} + 8660520 x^{3} y^{8} + 220 x^{2} ({(- 3)}^{9} y^{9}) + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.69

Rewrite using the commutative property of multiplication.

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} - 11547360 x^{4} y^{7} + 8660520 x^{3} y^{8} + 220 \cdot {(- 3)}^{9} x^{2} y^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.70

Raise $- 3$ to the power of $9$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} - 11547360 x^{4} y^{7} + 8660520 x^{3} y^{8} + 220 \cdot - 19683 x^{2} y^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.71

Multiply $220$ by $- 19683$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} - 11547360 x^{4} y^{7} + 8660520 x^{3} y^{8} - 4330260 x^{2} y^{9} + 11 {(2 x)}^{1} {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.72

Simplify.

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} - 11547360 x^{4} y^{7} + 8660520 x^{3} y^{8} - 4330260 x^{2} y^{9} + 11 (2 x) {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.73

Multiply $2$ by $11$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} - 11547360 x^{4} y^{7} + 8660520 x^{3} y^{8} - 4330260 x^{2} y^{9} + 22 x {(- 3 y)}^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.74

Apply the product rule to $- 3 y$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} - 11547360 x^{4} y^{7} + 8660520 x^{3} y^{8} - 4330260 x^{2} y^{9} + 22 x ({(- 3)}^{10} y^{10}) + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.75

Rewrite using the commutative property of multiplication.

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} - 11547360 x^{4} y^{7} + 8660520 x^{3} y^{8} - 4330260 x^{2} y^{9} + 22 \cdot {(- 3)}^{10} x y^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.76

Raise $- 3$ to the power of $10$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} - 11547360 x^{4} y^{7} + 8660520 x^{3} y^{8} - 4330260 x^{2} y^{9} + 22 \cdot 59049 x y^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.77

Multiply $22$ by $59049$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} - 11547360 x^{4} y^{7} + 8660520 x^{3} y^{8} - 4330260 x^{2} y^{9} + 1299078 x y^{10} + 1 {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.78

Multiply ${(2 x)}^{0}$ by $1$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} - 11547360 x^{4} y^{7} + 8660520 x^{3} y^{8} - 4330260 x^{2} y^{9} + 1299078 x y^{10} + {(2 x)}^{0} {(- 3 y)}^{11}$

Step 4.79

Apply the product rule to $2 x$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} - 11547360 x^{4} y^{7} + 8660520 x^{3} y^{8} - 4330260 x^{2} y^{9} + 1299078 x y^{10} + 2^{0} x^{0} {(- 3 y)}^{11}$

Step 4.80

Anything raised to $0$ is $1$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} - 11547360 x^{4} y^{7} + 8660520 x^{3} y^{8} - 4330260 x^{2} y^{9} + 1299078 x y^{10} + 1 x^{0} {(- 3 y)}^{11}$

Step 4.81

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

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} - 11547360 x^{4} y^{7} + 8660520 x^{3} y^{8} - 4330260 x^{2} y^{9} + 1299078 x y^{10} + x^{0} {(- 3 y)}^{11}$

Step 4.82

Anything raised to $0$ is $1$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} - 11547360 x^{4} y^{7} + 8660520 x^{3} y^{8} - 4330260 x^{2} y^{9} + 1299078 x y^{10} + 1 {(- 3 y)}^{11}$

Step 4.83

Multiply ${(- 3 y)}^{11}$ by $1$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} - 11547360 x^{4} y^{7} + 8660520 x^{3} y^{8} - 4330260 x^{2} y^{9} + 1299078 x y^{10} + {(- 3 y)}^{11}$

Step 4.84

Apply the product rule to $- 3 y$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} - 11547360 x^{4} y^{7} + 8660520 x^{3} y^{8} - 4330260 x^{2} y^{9} + 1299078 x y^{10} + {(- 3)}^{11} y^{11}$

Step 4.85

Raise $- 3$ to the power of $11$ .

$2048 x^{11} - 33792 x^{10} y + 253440 x^{9} y^{2} - 1140480 x^{8} y^{3} + 3421440 x^{7} y^{4} - 7185024 x^{6} y^{5} + 10777536 x^{5} y^{6} - 11547360 x^{4} y^{7} + 8660520 x^{3} y^{8} - 4330260 x^{2} y^{9} + 1299078 x y^{10} - 177147 y^{11}$