Expand Using Pascal's Triangle (2x-y)^9

${(2 x - y)}^{9}$

Step 1

Pascal\'s Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$\dots$

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

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

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

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 ${(2 x - y)}^{9}$ , $n = 9$ so the coefficients of the expansion will correspond with line $10$ .

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

$1 a^{9} b^{0} + 9 a^{8} b + 36 a^{7} b^{2} + 84 a^{6} b^{3} + 126 a^{5} b^{4} + 126 a^{4} b^{5} + 84 a^{3} b^{6} + 36 a^{2} b^{7} + 9 a b^{8} + 1 a^{0} b^{9}$

Step 3

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

$1 {(2 x)}^{9} {(- y)}^{0} + 9 {(2 x)}^{8} {(- y)}^{1} + 36 {(2 x)}^{7} {(- y)}^{2} + 84 {(2 x)}^{6} {(- y)}^{3} + 126 {(2 x)}^{5} {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Step 4

Simplify each term.

Multiply ${(2 x)}^{9}$ by $1$ .

${(2 x)}^{9} {(- y)}^{0} + 9 {(2 x)}^{8} {(- y)}^{1} + 36 {(2 x)}^{7} {(- y)}^{2} + 84 {(2 x)}^{6} {(- y)}^{3} + 126 {(2 x)}^{5} {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Apply the product rule to $2 x$ .

$2^{9} x^{9} {(- y)}^{0} + 9 {(2 x)}^{8} {(- y)}^{1} + 36 {(2 x)}^{7} {(- y)}^{2} + 84 {(2 x)}^{6} {(- y)}^{3} + 126 {(2 x)}^{5} {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

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

$512 x^{9} {(- y)}^{0} + 9 {(2 x)}^{8} {(- y)}^{1} + 36 {(2 x)}^{7} {(- y)}^{2} + 84 {(2 x)}^{6} {(- y)}^{3} + 126 {(2 x)}^{5} {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Apply the product rule to $- y$ .

$512 x^{9} ({(- 1)}^{0} y^{0}) + 9 {(2 x)}^{8} {(- y)}^{1} + 36 {(2 x)}^{7} {(- y)}^{2} + 84 {(2 x)}^{6} {(- y)}^{3} + 126 {(2 x)}^{5} {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Rewrite using the commutative property of multiplication.

$512 \cdot {(- 1)}^{0} x^{9} y^{0} + 9 {(2 x)}^{8} {(- y)}^{1} + 36 {(2 x)}^{7} {(- y)}^{2} + 84 {(2 x)}^{6} {(- y)}^{3} + 126 {(2 x)}^{5} {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Anything raised to $0$ is $1$ .

$512 \cdot 1 x^{9} y^{0} + 9 {(2 x)}^{8} {(- y)}^{1} + 36 {(2 x)}^{7} {(- y)}^{2} + 84 {(2 x)}^{6} {(- y)}^{3} + 126 {(2 x)}^{5} {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Multiply $512$ by $1$ .

$512 x^{9} y^{0} + 9 {(2 x)}^{8} {(- y)}^{1} + 36 {(2 x)}^{7} {(- y)}^{2} + 84 {(2 x)}^{6} {(- y)}^{3} + 126 {(2 x)}^{5} {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Anything raised to $0$ is $1$ .

$512 x^{9} \cdot 1 + 9 {(2 x)}^{8} {(- y)}^{1} + 36 {(2 x)}^{7} {(- y)}^{2} + 84 {(2 x)}^{6} {(- y)}^{3} + 126 {(2 x)}^{5} {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Multiply $512$ by $1$ .

$512 x^{9} + 9 {(2 x)}^{8} {(- y)}^{1} + 36 {(2 x)}^{7} {(- y)}^{2} + 84 {(2 x)}^{6} {(- y)}^{3} + 126 {(2 x)}^{5} {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Apply the product rule to $2 x$ .

$512 x^{9} + 9 (2^{8} x^{8}) {(- y)}^{1} + 36 {(2 x)}^{7} {(- y)}^{2} + 84 {(2 x)}^{6} {(- y)}^{3} + 126 {(2 x)}^{5} {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

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

$512 x^{9} + 9 (256 x^{8}) {(- y)}^{1} + 36 {(2 x)}^{7} {(- y)}^{2} + 84 {(2 x)}^{6} {(- y)}^{3} + 126 {(2 x)}^{5} {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Multiply $256$ by $9$ .

$512 x^{9} + 2304 x^{8} {(- y)}^{1} + 36 {(2 x)}^{7} {(- y)}^{2} + 84 {(2 x)}^{6} {(- y)}^{3} + 126 {(2 x)}^{5} {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Simplify.

$512 x^{9} + 2304 x^{8} (- y) + 36 {(2 x)}^{7} {(- y)}^{2} + 84 {(2 x)}^{6} {(- y)}^{3} + 126 {(2 x)}^{5} {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Rewrite using the commutative property of multiplication.

$512 x^{9} + 2304 \cdot - 1 x^{8} y + 36 {(2 x)}^{7} {(- y)}^{2} + 84 {(2 x)}^{6} {(- y)}^{3} + 126 {(2 x)}^{5} {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Multiply $2304$ by $- 1$ .

$512 x^{9} - 2304 x^{8} y + 36 {(2 x)}^{7} {(- y)}^{2} + 84 {(2 x)}^{6} {(- y)}^{3} + 126 {(2 x)}^{5} {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Apply the product rule to $2 x$ .

$512 x^{9} - 2304 x^{8} y + 36 (2^{7} x^{7}) {(- y)}^{2} + 84 {(2 x)}^{6} {(- y)}^{3} + 126 {(2 x)}^{5} {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

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

$512 x^{9} - 2304 x^{8} y + 36 (128 x^{7}) {(- y)}^{2} + 84 {(2 x)}^{6} {(- y)}^{3} + 126 {(2 x)}^{5} {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Multiply $128$ by $36$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} {(- y)}^{2} + 84 {(2 x)}^{6} {(- y)}^{3} + 126 {(2 x)}^{5} {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Apply the product rule to $- y$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} ({(- 1)}^{2} y^{2}) + 84 {(2 x)}^{6} {(- y)}^{3} + 126 {(2 x)}^{5} {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Rewrite using the commutative property of multiplication.

$512 x^{9} - 2304 x^{8} y + 4608 \cdot {(- 1)}^{2} x^{7} y^{2} + 84 {(2 x)}^{6} {(- y)}^{3} + 126 {(2 x)}^{5} {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

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

$512 x^{9} - 2304 x^{8} y + 4608 \cdot 1 x^{7} y^{2} + 84 {(2 x)}^{6} {(- y)}^{3} + 126 {(2 x)}^{5} {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Multiply $4608$ by $1$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} + 84 {(2 x)}^{6} {(- y)}^{3} + 126 {(2 x)}^{5} {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Apply the product rule to $2 x$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} + 84 (2^{6} x^{6}) {(- y)}^{3} + 126 {(2 x)}^{5} {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

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

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} + 84 (64 x^{6}) {(- y)}^{3} + 126 {(2 x)}^{5} {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Multiply $64$ by $84$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} + 5376 x^{6} {(- y)}^{3} + 126 {(2 x)}^{5} {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Apply the product rule to $- y$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} + 5376 x^{6} ({(- 1)}^{3} y^{3}) + 126 {(2 x)}^{5} {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Rewrite using the commutative property of multiplication.

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} + 5376 \cdot {(- 1)}^{3} x^{6} y^{3} + 126 {(2 x)}^{5} {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

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

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} + 5376 \cdot - 1 x^{6} y^{3} + 126 {(2 x)}^{5} {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Multiply $5376$ by $- 1$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 126 {(2 x)}^{5} {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Apply the product rule to $2 x$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 126 (2^{5} x^{5}) {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

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

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 126 (32 x^{5}) {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Multiply $32$ by $126$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} {(- y)}^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Apply the product rule to $- y$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} ({(- 1)}^{4} y^{4}) + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Rewrite using the commutative property of multiplication.

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 \cdot {(- 1)}^{4} x^{5} y^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Raise $- 1$ to the power of $4$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 \cdot 1 x^{5} y^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Multiply $4032$ by $1$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} + 126 {(2 x)}^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Apply the product rule to $2 x$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} + 126 (2^{4} x^{4}) {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

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

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} + 126 (16 x^{4}) {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Multiply $16$ by $126$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} + 2016 x^{4} {(- y)}^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Apply the product rule to $- y$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} + 2016 x^{4} ({(- 1)}^{5} y^{5}) + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Rewrite using the commutative property of multiplication.

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} + 2016 \cdot {(- 1)}^{5} x^{4} y^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Raise $- 1$ to the power of $5$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} + 2016 \cdot - 1 x^{4} y^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Multiply $2016$ by $- 1$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} - 2016 x^{4} y^{5} + 84 {(2 x)}^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Apply the product rule to $2 x$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} - 2016 x^{4} y^{5} + 84 (2^{3} x^{3}) {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

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

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} - 2016 x^{4} y^{5} + 84 (8 x^{3}) {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Multiply $8$ by $84$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} - 2016 x^{4} y^{5} + 672 x^{3} {(- y)}^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Apply the product rule to $- y$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} - 2016 x^{4} y^{5} + 672 x^{3} ({(- 1)}^{6} y^{6}) + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Rewrite using the commutative property of multiplication.

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} - 2016 x^{4} y^{5} + 672 \cdot {(- 1)}^{6} x^{3} y^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Raise $- 1$ to the power of $6$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} - 2016 x^{4} y^{5} + 672 \cdot 1 x^{3} y^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Multiply $672$ by $1$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} - 2016 x^{4} y^{5} + 672 x^{3} y^{6} + 36 {(2 x)}^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Apply the product rule to $2 x$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} - 2016 x^{4} y^{5} + 672 x^{3} y^{6} + 36 (2^{2} x^{2}) {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

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

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} - 2016 x^{4} y^{5} + 672 x^{3} y^{6} + 36 (4 x^{2}) {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Multiply $4$ by $36$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} - 2016 x^{4} y^{5} + 672 x^{3} y^{6} + 144 x^{2} {(- y)}^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Apply the product rule to $- y$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} - 2016 x^{4} y^{5} + 672 x^{3} y^{6} + 144 x^{2} ({(- 1)}^{7} y^{7}) + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Rewrite using the commutative property of multiplication.

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} - 2016 x^{4} y^{5} + 672 x^{3} y^{6} + 144 \cdot {(- 1)}^{7} x^{2} y^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Raise $- 1$ to the power of $7$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} - 2016 x^{4} y^{5} + 672 x^{3} y^{6} + 144 \cdot - 1 x^{2} y^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Multiply $144$ by $- 1$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} - 2016 x^{4} y^{5} + 672 x^{3} y^{6} - 144 x^{2} y^{7} + 9 {(2 x)}^{1} {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Simplify.

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} - 2016 x^{4} y^{5} + 672 x^{3} y^{6} - 144 x^{2} y^{7} + 9 (2 x) {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Multiply $2$ by $9$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} - 2016 x^{4} y^{5} + 672 x^{3} y^{6} - 144 x^{2} y^{7} + 18 x {(- y)}^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Apply the product rule to $- y$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} - 2016 x^{4} y^{5} + 672 x^{3} y^{6} - 144 x^{2} y^{7} + 18 x ({(- 1)}^{8} y^{8}) + 1 {(2 x)}^{0} {(- y)}^{9}$

Rewrite using the commutative property of multiplication.

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} - 2016 x^{4} y^{5} + 672 x^{3} y^{6} - 144 x^{2} y^{7} + 18 \cdot {(- 1)}^{8} x y^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Raise $- 1$ to the power of $8$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} - 2016 x^{4} y^{5} + 672 x^{3} y^{6} - 144 x^{2} y^{7} + 18 \cdot 1 x y^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

Multiply $18$ by $1$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} - 2016 x^{4} y^{5} + 672 x^{3} y^{6} - 144 x^{2} y^{7} + 18 x y^{8} + 1 {(2 x)}^{0} {(- y)}^{9}$

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

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} - 2016 x^{4} y^{5} + 672 x^{3} y^{6} - 144 x^{2} y^{7} + 18 x y^{8} + {(2 x)}^{0} {(- y)}^{9}$

Apply the product rule to $2 x$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} - 2016 x^{4} y^{5} + 672 x^{3} y^{6} - 144 x^{2} y^{7} + 18 x y^{8} + 2^{0} x^{0} {(- y)}^{9}$

Anything raised to $0$ is $1$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} - 2016 x^{4} y^{5} + 672 x^{3} y^{6} - 144 x^{2} y^{7} + 18 x y^{8} + 1 x^{0} {(- y)}^{9}$

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

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} - 2016 x^{4} y^{5} + 672 x^{3} y^{6} - 144 x^{2} y^{7} + 18 x y^{8} + x^{0} {(- y)}^{9}$

Anything raised to $0$ is $1$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} - 2016 x^{4} y^{5} + 672 x^{3} y^{6} - 144 x^{2} y^{7} + 18 x y^{8} + 1 {(- y)}^{9}$

Multiply ${(- y)}^{9}$ by $1$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} - 2016 x^{4} y^{5} + 672 x^{3} y^{6} - 144 x^{2} y^{7} + 18 x y^{8} + {(- y)}^{9}$

Apply the product rule to $- y$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} - 2016 x^{4} y^{5} + 672 x^{3} y^{6} - 144 x^{2} y^{7} + 18 x y^{8} + {(- 1)}^{9} y^{9}$

Raise $- 1$ to the power of $9$ .

$512 x^{9} - 2304 x^{8} y + 4608 x^{7} y^{2} - 5376 x^{6} y^{3} + 4032 x^{5} y^{4} - 2016 x^{4} y^{5} + 672 x^{3} y^{6} - 144 x^{2} y^{7} + 18 x y^{8} - y^{9}$

Expand Using Pascal's Triangle (2x-y)^9

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