Expand Using Pascal's Triangle (3x-3y)^7

${(3 x - 3 y)}^{7}$

Step 1

Pascal\'s Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$\dots$

$1 - 5 - 10 - 10 - 5 - 1$

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

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

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 ${(3 x - 3 y)}^{7}$ , $n = 7$ so the coefficients of the expansion will correspond with line $8$ .

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

$1 a^{7} b^{0} + 7 a^{6} b + 21 a^{5} b^{2} + 35 a^{4} b^{3} + 35 a^{3} b^{4} + 21 a^{2} b^{5} + 7 a b^{6} + 1 a^{0} b^{7}$

Step 3

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

$1 {(3 x)}^{7} {(- 3 y)}^{0} + 7 {(3 x)}^{6} {(- 3 y)}^{1} + 21 {(3 x)}^{5} {(- 3 y)}^{2} + 35 {(3 x)}^{4} {(- 3 y)}^{3} + 35 {(3 x)}^{3} {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Step 4

Simplify each term.

Multiply ${(3 x)}^{7}$ by $1$ .

${(3 x)}^{7} {(- 3 y)}^{0} + 7 {(3 x)}^{6} {(- 3 y)}^{1} + 21 {(3 x)}^{5} {(- 3 y)}^{2} + 35 {(3 x)}^{4} {(- 3 y)}^{3} + 35 {(3 x)}^{3} {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Apply the product rule to $3 x$ .

$3^{7} x^{7} {(- 3 y)}^{0} + 7 {(3 x)}^{6} {(- 3 y)}^{1} + 21 {(3 x)}^{5} {(- 3 y)}^{2} + 35 {(3 x)}^{4} {(- 3 y)}^{3} + 35 {(3 x)}^{3} {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

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

$2187 x^{7} {(- 3 y)}^{0} + 7 {(3 x)}^{6} {(- 3 y)}^{1} + 21 {(3 x)}^{5} {(- 3 y)}^{2} + 35 {(3 x)}^{4} {(- 3 y)}^{3} + 35 {(3 x)}^{3} {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Apply the product rule to $- 3 y$ .

$2187 x^{7} ({(- 3)}^{0} y^{0}) + 7 {(3 x)}^{6} {(- 3 y)}^{1} + 21 {(3 x)}^{5} {(- 3 y)}^{2} + 35 {(3 x)}^{4} {(- 3 y)}^{3} + 35 {(3 x)}^{3} {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Rewrite using the commutative property of multiplication.

$2187 \cdot {(- 3)}^{0} x^{7} y^{0} + 7 {(3 x)}^{6} {(- 3 y)}^{1} + 21 {(3 x)}^{5} {(- 3 y)}^{2} + 35 {(3 x)}^{4} {(- 3 y)}^{3} + 35 {(3 x)}^{3} {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Anything raised to $0$ is $1$ .

$2187 \cdot 1 x^{7} y^{0} + 7 {(3 x)}^{6} {(- 3 y)}^{1} + 21 {(3 x)}^{5} {(- 3 y)}^{2} + 35 {(3 x)}^{4} {(- 3 y)}^{3} + 35 {(3 x)}^{3} {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Multiply $2187$ by $1$ .

$2187 x^{7} y^{0} + 7 {(3 x)}^{6} {(- 3 y)}^{1} + 21 {(3 x)}^{5} {(- 3 y)}^{2} + 35 {(3 x)}^{4} {(- 3 y)}^{3} + 35 {(3 x)}^{3} {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Anything raised to $0$ is $1$ .

$2187 x^{7} \cdot 1 + 7 {(3 x)}^{6} {(- 3 y)}^{1} + 21 {(3 x)}^{5} {(- 3 y)}^{2} + 35 {(3 x)}^{4} {(- 3 y)}^{3} + 35 {(3 x)}^{3} {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Multiply $2187$ by $1$ .

$2187 x^{7} + 7 {(3 x)}^{6} {(- 3 y)}^{1} + 21 {(3 x)}^{5} {(- 3 y)}^{2} + 35 {(3 x)}^{4} {(- 3 y)}^{3} + 35 {(3 x)}^{3} {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Apply the product rule to $3 x$ .

$2187 x^{7} + 7 (3^{6} x^{6}) {(- 3 y)}^{1} + 21 {(3 x)}^{5} {(- 3 y)}^{2} + 35 {(3 x)}^{4} {(- 3 y)}^{3} + 35 {(3 x)}^{3} {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

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

$2187 x^{7} + 7 (729 x^{6}) {(- 3 y)}^{1} + 21 {(3 x)}^{5} {(- 3 y)}^{2} + 35 {(3 x)}^{4} {(- 3 y)}^{3} + 35 {(3 x)}^{3} {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Multiply $729$ by $7$ .

$2187 x^{7} + 5103 x^{6} {(- 3 y)}^{1} + 21 {(3 x)}^{5} {(- 3 y)}^{2} + 35 {(3 x)}^{4} {(- 3 y)}^{3} + 35 {(3 x)}^{3} {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Simplify.

$2187 x^{7} + 5103 x^{6} (- 3 y) + 21 {(3 x)}^{5} {(- 3 y)}^{2} + 35 {(3 x)}^{4} {(- 3 y)}^{3} + 35 {(3 x)}^{3} {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Rewrite using the commutative property of multiplication.

$2187 x^{7} + 5103 \cdot - 3 x^{6} y + 21 {(3 x)}^{5} {(- 3 y)}^{2} + 35 {(3 x)}^{4} {(- 3 y)}^{3} + 35 {(3 x)}^{3} {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Multiply $5103$ by $- 3$ .

$2187 x^{7} - 15309 x^{6} y + 21 {(3 x)}^{5} {(- 3 y)}^{2} + 35 {(3 x)}^{4} {(- 3 y)}^{3} + 35 {(3 x)}^{3} {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Apply the product rule to $3 x$ .

$2187 x^{7} - 15309 x^{6} y + 21 (3^{5} x^{5}) {(- 3 y)}^{2} + 35 {(3 x)}^{4} {(- 3 y)}^{3} + 35 {(3 x)}^{3} {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

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

$2187 x^{7} - 15309 x^{6} y + 21 (243 x^{5}) {(- 3 y)}^{2} + 35 {(3 x)}^{4} {(- 3 y)}^{3} + 35 {(3 x)}^{3} {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Multiply $243$ by $21$ .

$2187 x^{7} - 15309 x^{6} y + 5103 x^{5} {(- 3 y)}^{2} + 35 {(3 x)}^{4} {(- 3 y)}^{3} + 35 {(3 x)}^{3} {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Apply the product rule to $- 3 y$ .

$2187 x^{7} - 15309 x^{6} y + 5103 x^{5} ({(- 3)}^{2} y^{2}) + 35 {(3 x)}^{4} {(- 3 y)}^{3} + 35 {(3 x)}^{3} {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Rewrite using the commutative property of multiplication.

$2187 x^{7} - 15309 x^{6} y + 5103 \cdot {(- 3)}^{2} x^{5} y^{2} + 35 {(3 x)}^{4} {(- 3 y)}^{3} + 35 {(3 x)}^{3} {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

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

$2187 x^{7} - 15309 x^{6} y + 5103 \cdot 9 x^{5} y^{2} + 35 {(3 x)}^{4} {(- 3 y)}^{3} + 35 {(3 x)}^{3} {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Multiply $5103$ by $9$ .

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} + 35 {(3 x)}^{4} {(- 3 y)}^{3} + 35 {(3 x)}^{3} {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Apply the product rule to $3 x$ .

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} + 35 (3^{4} x^{4}) {(- 3 y)}^{3} + 35 {(3 x)}^{3} {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

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

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} + 35 (81 x^{4}) {(- 3 y)}^{3} + 35 {(3 x)}^{3} {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Multiply $81$ by $35$ .

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} + 2835 x^{4} {(- 3 y)}^{3} + 35 {(3 x)}^{3} {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Apply the product rule to $- 3 y$ .

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} + 2835 x^{4} ({(- 3)}^{3} y^{3}) + 35 {(3 x)}^{3} {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Rewrite using the commutative property of multiplication.

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} + 2835 \cdot {(- 3)}^{3} x^{4} y^{3} + 35 {(3 x)}^{3} {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

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

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} + 2835 \cdot - 27 x^{4} y^{3} + 35 {(3 x)}^{3} {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Multiply $2835$ by $- 27$ .

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} - 76545 x^{4} y^{3} + 35 {(3 x)}^{3} {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Apply the product rule to $3 x$ .

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} - 76545 x^{4} y^{3} + 35 (3^{3} x^{3}) {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

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

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} - 76545 x^{4} y^{3} + 35 (27 x^{3}) {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Multiply $27$ by $35$ .

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} - 76545 x^{4} y^{3} + 945 x^{3} {(- 3 y)}^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Apply the product rule to $- 3 y$ .

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} - 76545 x^{4} y^{3} + 945 x^{3} ({(- 3)}^{4} y^{4}) + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Rewrite using the commutative property of multiplication.

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} - 76545 x^{4} y^{3} + 945 \cdot {(- 3)}^{4} x^{3} y^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

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

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} - 76545 x^{4} y^{3} + 945 \cdot 81 x^{3} y^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Multiply $945$ by $81$ .

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} - 76545 x^{4} y^{3} + 76545 x^{3} y^{4} + 21 {(3 x)}^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Apply the product rule to $3 x$ .

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} - 76545 x^{4} y^{3} + 76545 x^{3} y^{4} + 21 (3^{2} x^{2}) {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

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

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} - 76545 x^{4} y^{3} + 76545 x^{3} y^{4} + 21 (9 x^{2}) {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Multiply $9$ by $21$ .

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} - 76545 x^{4} y^{3} + 76545 x^{3} y^{4} + 189 x^{2} {(- 3 y)}^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Apply the product rule to $- 3 y$ .

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} - 76545 x^{4} y^{3} + 76545 x^{3} y^{4} + 189 x^{2} ({(- 3)}^{5} y^{5}) + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Rewrite using the commutative property of multiplication.

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} - 76545 x^{4} y^{3} + 76545 x^{3} y^{4} + 189 \cdot {(- 3)}^{5} x^{2} y^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

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

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} - 76545 x^{4} y^{3} + 76545 x^{3} y^{4} + 189 \cdot - 243 x^{2} y^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Multiply $189$ by $- 243$ .

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} - 76545 x^{4} y^{3} + 76545 x^{3} y^{4} - 45927 x^{2} y^{5} + 7 {(3 x)}^{1} {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Simplify.

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} - 76545 x^{4} y^{3} + 76545 x^{3} y^{4} - 45927 x^{2} y^{5} + 7 (3 x) {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Multiply $3$ by $7$ .

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} - 76545 x^{4} y^{3} + 76545 x^{3} y^{4} - 45927 x^{2} y^{5} + 21 x {(- 3 y)}^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Apply the product rule to $- 3 y$ .

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} - 76545 x^{4} y^{3} + 76545 x^{3} y^{4} - 45927 x^{2} y^{5} + 21 x ({(- 3)}^{6} y^{6}) + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Rewrite using the commutative property of multiplication.

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} - 76545 x^{4} y^{3} + 76545 x^{3} y^{4} - 45927 x^{2} y^{5} + 21 \cdot {(- 3)}^{6} x y^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

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

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} - 76545 x^{4} y^{3} + 76545 x^{3} y^{4} - 45927 x^{2} y^{5} + 21 \cdot 729 x y^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

Multiply $21$ by $729$ .

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} - 76545 x^{4} y^{3} + 76545 x^{3} y^{4} - 45927 x^{2} y^{5} + 15309 x y^{6} + 1 {(3 x)}^{0} {(- 3 y)}^{7}$

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

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} - 76545 x^{4} y^{3} + 76545 x^{3} y^{4} - 45927 x^{2} y^{5} + 15309 x y^{6} + {(3 x)}^{0} {(- 3 y)}^{7}$

Apply the product rule to $3 x$ .

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} - 76545 x^{4} y^{3} + 76545 x^{3} y^{4} - 45927 x^{2} y^{5} + 15309 x y^{6} + 3^{0} x^{0} {(- 3 y)}^{7}$

Anything raised to $0$ is $1$ .

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} - 76545 x^{4} y^{3} + 76545 x^{3} y^{4} - 45927 x^{2} y^{5} + 15309 x y^{6} + 1 x^{0} {(- 3 y)}^{7}$

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

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} - 76545 x^{4} y^{3} + 76545 x^{3} y^{4} - 45927 x^{2} y^{5} + 15309 x y^{6} + x^{0} {(- 3 y)}^{7}$

Anything raised to $0$ is $1$ .

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} - 76545 x^{4} y^{3} + 76545 x^{3} y^{4} - 45927 x^{2} y^{5} + 15309 x y^{6} + 1 {(- 3 y)}^{7}$

Multiply ${(- 3 y)}^{7}$ by $1$ .

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} - 76545 x^{4} y^{3} + 76545 x^{3} y^{4} - 45927 x^{2} y^{5} + 15309 x y^{6} + {(- 3 y)}^{7}$

Apply the product rule to $- 3 y$ .

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} - 76545 x^{4} y^{3} + 76545 x^{3} y^{4} - 45927 x^{2} y^{5} + 15309 x y^{6} + {(- 3)}^{7} y^{7}$

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

$2187 x^{7} - 15309 x^{6} y + 45927 x^{5} y^{2} - 76545 x^{4} y^{3} + 76545 x^{3} y^{4} - 45927 x^{2} y^{5} + 15309 x y^{6} - 2187 y^{7}$

Expand Using Pascal's Triangle (3x-3y)^7

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