Expand the Trigonometric Expression (2x-3y)^7

${(2 x - 3 y)}^{7}$

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify each term.

Apply the product rule to $2 x$ .

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

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

$128 x^{7} + 7 {(2 x)}^{6} (- 3 y) + 21 {(2 x)}^{5} {(- 3 y)}^{2} + 35 {(2 x)}^{4} {(- 3 y)}^{3} + 35 {(2 x)}^{3} {(- 3 y)}^{4} + 21 {(2 x)}^{2} {(- 3 y)}^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

Rewrite using the commutative property of multiplication.

$128 x^{7} + 7 \cdot - 3 {(2 x)}^{6} y + 21 {(2 x)}^{5} {(- 3 y)}^{2} + 35 {(2 x)}^{4} {(- 3 y)}^{3} + 35 {(2 x)}^{3} {(- 3 y)}^{4} + 21 {(2 x)}^{2} {(- 3 y)}^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

Multiply $7$ by $- 3$ .

$128 x^{7} - 21 {(2 x)}^{6} y + 21 {(2 x)}^{5} {(- 3 y)}^{2} + 35 {(2 x)}^{4} {(- 3 y)}^{3} + 35 {(2 x)}^{3} {(- 3 y)}^{4} + 21 {(2 x)}^{2} {(- 3 y)}^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

Apply the product rule to $2 x$ .

$128 x^{7} - 21 (2^{6} x^{6}) y + 21 {(2 x)}^{5} {(- 3 y)}^{2} + 35 {(2 x)}^{4} {(- 3 y)}^{3} + 35 {(2 x)}^{3} {(- 3 y)}^{4} + 21 {(2 x)}^{2} {(- 3 y)}^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

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

$128 x^{7} - 21 (64 x^{6}) y + 21 {(2 x)}^{5} {(- 3 y)}^{2} + 35 {(2 x)}^{4} {(- 3 y)}^{3} + 35 {(2 x)}^{3} {(- 3 y)}^{4} + 21 {(2 x)}^{2} {(- 3 y)}^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

Multiply $64$ by $- 21$ .

$128 x^{7} - 1344 x^{6} y + 21 {(2 x)}^{5} {(- 3 y)}^{2} + 35 {(2 x)}^{4} {(- 3 y)}^{3} + 35 {(2 x)}^{3} {(- 3 y)}^{4} + 21 {(2 x)}^{2} {(- 3 y)}^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

Apply the product rule to $2 x$ .

$128 x^{7} - 1344 x^{6} y + 21 (2^{5} x^{5}) {(- 3 y)}^{2} + 35 {(2 x)}^{4} {(- 3 y)}^{3} + 35 {(2 x)}^{3} {(- 3 y)}^{4} + 21 {(2 x)}^{2} {(- 3 y)}^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

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

$128 x^{7} - 1344 x^{6} y + 21 (32 x^{5}) {(- 3 y)}^{2} + 35 {(2 x)}^{4} {(- 3 y)}^{3} + 35 {(2 x)}^{3} {(- 3 y)}^{4} + 21 {(2 x)}^{2} {(- 3 y)}^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

Multiply $32$ by $21$ .

$128 x^{7} - 1344 x^{6} y + 672 x^{5} {(- 3 y)}^{2} + 35 {(2 x)}^{4} {(- 3 y)}^{3} + 35 {(2 x)}^{3} {(- 3 y)}^{4} + 21 {(2 x)}^{2} {(- 3 y)}^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

Apply the product rule to $- 3 y$ .

$128 x^{7} - 1344 x^{6} y + 672 x^{5} ({(- 3)}^{2} y^{2}) + 35 {(2 x)}^{4} {(- 3 y)}^{3} + 35 {(2 x)}^{3} {(- 3 y)}^{4} + 21 {(2 x)}^{2} {(- 3 y)}^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

Rewrite using the commutative property of multiplication.

$128 x^{7} - 1344 x^{6} y + 672 \cdot {(- 3)}^{2} x^{5} y^{2} + 35 {(2 x)}^{4} {(- 3 y)}^{3} + 35 {(2 x)}^{3} {(- 3 y)}^{4} + 21 {(2 x)}^{2} {(- 3 y)}^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

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

$128 x^{7} - 1344 x^{6} y + 672 \cdot 9 x^{5} y^{2} + 35 {(2 x)}^{4} {(- 3 y)}^{3} + 35 {(2 x)}^{3} {(- 3 y)}^{4} + 21 {(2 x)}^{2} {(- 3 y)}^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

Multiply $672$ by $9$ .

$128 x^{7} - 1344 x^{6} y + 6048 x^{5} y^{2} + 35 {(2 x)}^{4} {(- 3 y)}^{3} + 35 {(2 x)}^{3} {(- 3 y)}^{4} + 21 {(2 x)}^{2} {(- 3 y)}^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

Apply the product rule to $2 x$ .

$128 x^{7} - 1344 x^{6} y + 6048 x^{5} y^{2} + 35 (2^{4} x^{4}) {(- 3 y)}^{3} + 35 {(2 x)}^{3} {(- 3 y)}^{4} + 21 {(2 x)}^{2} {(- 3 y)}^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

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

$128 x^{7} - 1344 x^{6} y + 6048 x^{5} y^{2} + 35 (16 x^{4}) {(- 3 y)}^{3} + 35 {(2 x)}^{3} {(- 3 y)}^{4} + 21 {(2 x)}^{2} {(- 3 y)}^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

Multiply $16$ by $35$ .

$128 x^{7} - 1344 x^{6} y + 6048 x^{5} y^{2} + 560 x^{4} {(- 3 y)}^{3} + 35 {(2 x)}^{3} {(- 3 y)}^{4} + 21 {(2 x)}^{2} {(- 3 y)}^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

Apply the product rule to $- 3 y$ .

$128 x^{7} - 1344 x^{6} y + 6048 x^{5} y^{2} + 560 x^{4} ({(- 3)}^{3} y^{3}) + 35 {(2 x)}^{3} {(- 3 y)}^{4} + 21 {(2 x)}^{2} {(- 3 y)}^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

Rewrite using the commutative property of multiplication.

$128 x^{7} - 1344 x^{6} y + 6048 x^{5} y^{2} + 560 \cdot {(- 3)}^{3} x^{4} y^{3} + 35 {(2 x)}^{3} {(- 3 y)}^{4} + 21 {(2 x)}^{2} {(- 3 y)}^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

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

$128 x^{7} - 1344 x^{6} y + 6048 x^{5} y^{2} + 560 \cdot - 27 x^{4} y^{3} + 35 {(2 x)}^{3} {(- 3 y)}^{4} + 21 {(2 x)}^{2} {(- 3 y)}^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

Multiply $560$ by $- 27$ .

$128 x^{7} - 1344 x^{6} y + 6048 x^{5} y^{2} - 15120 x^{4} y^{3} + 35 {(2 x)}^{3} {(- 3 y)}^{4} + 21 {(2 x)}^{2} {(- 3 y)}^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

Apply the product rule to $2 x$ .

$128 x^{7} - 1344 x^{6} y + 6048 x^{5} y^{2} - 15120 x^{4} y^{3} + 35 (2^{3} x^{3}) {(- 3 y)}^{4} + 21 {(2 x)}^{2} {(- 3 y)}^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

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

$128 x^{7} - 1344 x^{6} y + 6048 x^{5} y^{2} - 15120 x^{4} y^{3} + 35 (8 x^{3}) {(- 3 y)}^{4} + 21 {(2 x)}^{2} {(- 3 y)}^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

Multiply $8$ by $35$ .

$128 x^{7} - 1344 x^{6} y + 6048 x^{5} y^{2} - 15120 x^{4} y^{3} + 280 x^{3} {(- 3 y)}^{4} + 21 {(2 x)}^{2} {(- 3 y)}^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

Apply the product rule to $- 3 y$ .

$128 x^{7} - 1344 x^{6} y + 6048 x^{5} y^{2} - 15120 x^{4} y^{3} + 280 x^{3} ({(- 3)}^{4} y^{4}) + 21 {(2 x)}^{2} {(- 3 y)}^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

Rewrite using the commutative property of multiplication.

$128 x^{7} - 1344 x^{6} y + 6048 x^{5} y^{2} - 15120 x^{4} y^{3} + 280 \cdot {(- 3)}^{4} x^{3} y^{4} + 21 {(2 x)}^{2} {(- 3 y)}^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

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

$128 x^{7} - 1344 x^{6} y + 6048 x^{5} y^{2} - 15120 x^{4} y^{3} + 280 \cdot 81 x^{3} y^{4} + 21 {(2 x)}^{2} {(- 3 y)}^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

Multiply $280$ by $81$ .

$128 x^{7} - 1344 x^{6} y + 6048 x^{5} y^{2} - 15120 x^{4} y^{3} + 22680 x^{3} y^{4} + 21 {(2 x)}^{2} {(- 3 y)}^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

Apply the product rule to $2 x$ .

$128 x^{7} - 1344 x^{6} y + 6048 x^{5} y^{2} - 15120 x^{4} y^{3} + 22680 x^{3} y^{4} + 21 (2^{2} x^{2}) {(- 3 y)}^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

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

$128 x^{7} - 1344 x^{6} y + 6048 x^{5} y^{2} - 15120 x^{4} y^{3} + 22680 x^{3} y^{4} + 21 (4 x^{2}) {(- 3 y)}^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

Multiply $4$ by $21$ .

$128 x^{7} - 1344 x^{6} y + 6048 x^{5} y^{2} - 15120 x^{4} y^{3} + 22680 x^{3} y^{4} + 84 x^{2} {(- 3 y)}^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

Apply the product rule to $- 3 y$ .

$128 x^{7} - 1344 x^{6} y + 6048 x^{5} y^{2} - 15120 x^{4} y^{3} + 22680 x^{3} y^{4} + 84 x^{2} ({(- 3)}^{5} y^{5}) + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

Rewrite using the commutative property of multiplication.

$128 x^{7} - 1344 x^{6} y + 6048 x^{5} y^{2} - 15120 x^{4} y^{3} + 22680 x^{3} y^{4} + 84 \cdot {(- 3)}^{5} x^{2} y^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

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

$128 x^{7} - 1344 x^{6} y + 6048 x^{5} y^{2} - 15120 x^{4} y^{3} + 22680 x^{3} y^{4} + 84 \cdot - 243 x^{2} y^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

Multiply $84$ by $- 243$ .

$128 x^{7} - 1344 x^{6} y + 6048 x^{5} y^{2} - 15120 x^{4} y^{3} + 22680 x^{3} y^{4} - 20412 x^{2} y^{5} + 7 (2 x) {(- 3 y)}^{6} + {(- 3 y)}^{7}$

Multiply $2$ by $7$ .

$128 x^{7} - 1344 x^{6} y + 6048 x^{5} y^{2} - 15120 x^{4} y^{3} + 22680 x^{3} y^{4} - 20412 x^{2} y^{5} + 14 x {(- 3 y)}^{6} + {(- 3 y)}^{7}$

Apply the product rule to $- 3 y$ .

$128 x^{7} - 1344 x^{6} y + 6048 x^{5} y^{2} - 15120 x^{4} y^{3} + 22680 x^{3} y^{4} - 20412 x^{2} y^{5} + 14 x ({(- 3)}^{6} y^{6}) + {(- 3 y)}^{7}$

Rewrite using the commutative property of multiplication.

$128 x^{7} - 1344 x^{6} y + 6048 x^{5} y^{2} - 15120 x^{4} y^{3} + 22680 x^{3} y^{4} - 20412 x^{2} y^{5} + 14 \cdot {(- 3)}^{6} x y^{6} + {(- 3 y)}^{7}$

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

$128 x^{7} - 1344 x^{6} y + 6048 x^{5} y^{2} - 15120 x^{4} y^{3} + 22680 x^{3} y^{4} - 20412 x^{2} y^{5} + 14 \cdot 729 x y^{6} + {(- 3 y)}^{7}$

Multiply $14$ by $729$ .

$128 x^{7} - 1344 x^{6} y + 6048 x^{5} y^{2} - 15120 x^{4} y^{3} + 22680 x^{3} y^{4} - 20412 x^{2} y^{5} + 10206 x y^{6} + {(- 3 y)}^{7}$

Apply the product rule to $- 3 y$ .

$128 x^{7} - 1344 x^{6} y + 6048 x^{5} y^{2} - 15120 x^{4} y^{3} + 22680 x^{3} y^{4} - 20412 x^{2} y^{5} + 10206 x y^{6} + {(- 3)}^{7} y^{7}$

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

$128 x^{7} - 1344 x^{6} y + 6048 x^{5} y^{2} - 15120 x^{4} y^{3} + 22680 x^{3} y^{4} - 20412 x^{2} y^{5} + 10206 x y^{6} - 2187 y^{7}$

Expand the Trigonometric Expression (2x-3y)^7

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