Expand the Trigonometric Expression (3x-3y)^7

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

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify each term.

Apply the product rule to $3 x$ .

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

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

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

Rewrite using the commutative property of multiplication.

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

Multiply $7$ by $- 3$ .

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

Apply the product rule to $3 x$ .

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

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

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

Multiply $729$ by $- 21$ .

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

Apply the product rule to $3 x$ .

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

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

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

Multiply $243$ by $21$ .

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

Apply the product rule to $- 3 y$ .

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

Rewrite using the commutative property of multiplication.

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

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

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

Multiply $5103$ by $9$ .

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

Apply the product rule to $3 x$ .

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

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

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

Multiply $81$ by $35$ .

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

Apply the product rule to $- 3 y$ .

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

Rewrite using the commutative property of multiplication.

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

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

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

Multiply $2835$ by $- 27$ .

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

Apply the product rule to $3 x$ .

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

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

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

Multiply $27$ by $35$ .

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

Apply the product rule to $- 3 y$ .

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

Rewrite using the commutative property of multiplication.

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

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

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

Multiply $945$ by $81$ .

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

Apply the product rule to $3 x$ .

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

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

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

Multiply $9$ by $21$ .

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

Apply the product rule to $- 3 y$ .

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

Rewrite using the commutative property of multiplication.

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

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

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

Multiply $189$ by $- 243$ .

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

Multiply $3$ by $7$ .

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

Apply the product rule to $- 3 y$ .

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

Rewrite using the commutative property of multiplication.

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

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

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

Multiply $21$ by $729$ .

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

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 the Trigonometric Expression (3x-3y)^7

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