Expand the Trigonometric Expression (2x-3y)^5

${(2 x - 3 y)}^{5}$

Use the Binomial Theorem.

${(2 x)}^{5} + 5 {(2 x)}^{4} (- 3 y) + 10 {(2 x)}^{3} {(- 3 y)}^{2} + 10 {(2 x)}^{2} {(- 3 y)}^{3} + 5 (2 x) {(- 3 y)}^{4} + {(- 3 y)}^{5}$

Simplify each term.

Apply the product rule to $2 x$ .

$2^{5} x^{5} + 5 {(2 x)}^{4} (- 3 y) + 10 {(2 x)}^{3} {(- 3 y)}^{2} + 10 {(2 x)}^{2} {(- 3 y)}^{3} + 5 (2 x) {(- 3 y)}^{4} + {(- 3 y)}^{5}$

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

$32 x^{5} + 5 {(2 x)}^{4} (- 3 y) + 10 {(2 x)}^{3} {(- 3 y)}^{2} + 10 {(2 x)}^{2} {(- 3 y)}^{3} + 5 (2 x) {(- 3 y)}^{4} + {(- 3 y)}^{5}$

Rewrite using the commutative property of multiplication.

$32 x^{5} + 5 \cdot - 3 {(2 x)}^{4} y + 10 {(2 x)}^{3} {(- 3 y)}^{2} + 10 {(2 x)}^{2} {(- 3 y)}^{3} + 5 (2 x) {(- 3 y)}^{4} + {(- 3 y)}^{5}$

Multiply $5$ by $- 3$ .

$32 x^{5} - 15 {(2 x)}^{4} y + 10 {(2 x)}^{3} {(- 3 y)}^{2} + 10 {(2 x)}^{2} {(- 3 y)}^{3} + 5 (2 x) {(- 3 y)}^{4} + {(- 3 y)}^{5}$

Apply the product rule to $2 x$ .

$32 x^{5} - 15 (2^{4} x^{4}) y + 10 {(2 x)}^{3} {(- 3 y)}^{2} + 10 {(2 x)}^{2} {(- 3 y)}^{3} + 5 (2 x) {(- 3 y)}^{4} + {(- 3 y)}^{5}$

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

$32 x^{5} - 15 (16 x^{4}) y + 10 {(2 x)}^{3} {(- 3 y)}^{2} + 10 {(2 x)}^{2} {(- 3 y)}^{3} + 5 (2 x) {(- 3 y)}^{4} + {(- 3 y)}^{5}$

Multiply $16$ by $- 15$ .

$32 x^{5} - 240 x^{4} y + 10 {(2 x)}^{3} {(- 3 y)}^{2} + 10 {(2 x)}^{2} {(- 3 y)}^{3} + 5 (2 x) {(- 3 y)}^{4} + {(- 3 y)}^{5}$

Apply the product rule to $2 x$ .

$32 x^{5} - 240 x^{4} y + 10 (2^{3} x^{3}) {(- 3 y)}^{2} + 10 {(2 x)}^{2} {(- 3 y)}^{3} + 5 (2 x) {(- 3 y)}^{4} + {(- 3 y)}^{5}$

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

$32 x^{5} - 240 x^{4} y + 10 (8 x^{3}) {(- 3 y)}^{2} + 10 {(2 x)}^{2} {(- 3 y)}^{3} + 5 (2 x) {(- 3 y)}^{4} + {(- 3 y)}^{5}$

Multiply $8$ by $10$ .

$32 x^{5} - 240 x^{4} y + 80 x^{3} {(- 3 y)}^{2} + 10 {(2 x)}^{2} {(- 3 y)}^{3} + 5 (2 x) {(- 3 y)}^{4} + {(- 3 y)}^{5}$

Apply the product rule to $- 3 y$ .

$32 x^{5} - 240 x^{4} y + 80 x^{3} ({(- 3)}^{2} y^{2}) + 10 {(2 x)}^{2} {(- 3 y)}^{3} + 5 (2 x) {(- 3 y)}^{4} + {(- 3 y)}^{5}$

Rewrite using the commutative property of multiplication.

$32 x^{5} - 240 x^{4} y + 80 \cdot {(- 3)}^{2} x^{3} y^{2} + 10 {(2 x)}^{2} {(- 3 y)}^{3} + 5 (2 x) {(- 3 y)}^{4} + {(- 3 y)}^{5}$

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

$32 x^{5} - 240 x^{4} y + 80 \cdot 9 x^{3} y^{2} + 10 {(2 x)}^{2} {(- 3 y)}^{3} + 5 (2 x) {(- 3 y)}^{4} + {(- 3 y)}^{5}$

Multiply $80$ by $9$ .

$32 x^{5} - 240 x^{4} y + 720 x^{3} y^{2} + 10 {(2 x)}^{2} {(- 3 y)}^{3} + 5 (2 x) {(- 3 y)}^{4} + {(- 3 y)}^{5}$

Apply the product rule to $2 x$ .

$32 x^{5} - 240 x^{4} y + 720 x^{3} y^{2} + 10 (2^{2} x^{2}) {(- 3 y)}^{3} + 5 (2 x) {(- 3 y)}^{4} + {(- 3 y)}^{5}$

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

$32 x^{5} - 240 x^{4} y + 720 x^{3} y^{2} + 10 (4 x^{2}) {(- 3 y)}^{3} + 5 (2 x) {(- 3 y)}^{4} + {(- 3 y)}^{5}$

Multiply $4$ by $10$ .

$32 x^{5} - 240 x^{4} y + 720 x^{3} y^{2} + 40 x^{2} {(- 3 y)}^{3} + 5 (2 x) {(- 3 y)}^{4} + {(- 3 y)}^{5}$

Apply the product rule to $- 3 y$ .

$32 x^{5} - 240 x^{4} y + 720 x^{3} y^{2} + 40 x^{2} ({(- 3)}^{3} y^{3}) + 5 (2 x) {(- 3 y)}^{4} + {(- 3 y)}^{5}$

Rewrite using the commutative property of multiplication.

$32 x^{5} - 240 x^{4} y + 720 x^{3} y^{2} + 40 \cdot {(- 3)}^{3} x^{2} y^{3} + 5 (2 x) {(- 3 y)}^{4} + {(- 3 y)}^{5}$

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

$32 x^{5} - 240 x^{4} y + 720 x^{3} y^{2} + 40 \cdot - 27 x^{2} y^{3} + 5 (2 x) {(- 3 y)}^{4} + {(- 3 y)}^{5}$

Multiply $40$ by $- 27$ .

$32 x^{5} - 240 x^{4} y + 720 x^{3} y^{2} - 1080 x^{2} y^{3} + 5 (2 x) {(- 3 y)}^{4} + {(- 3 y)}^{5}$

Multiply $2$ by $5$ .

$32 x^{5} - 240 x^{4} y + 720 x^{3} y^{2} - 1080 x^{2} y^{3} + 10 x {(- 3 y)}^{4} + {(- 3 y)}^{5}$

Apply the product rule to $- 3 y$ .

$32 x^{5} - 240 x^{4} y + 720 x^{3} y^{2} - 1080 x^{2} y^{3} + 10 x ({(- 3)}^{4} y^{4}) + {(- 3 y)}^{5}$

Rewrite using the commutative property of multiplication.

$32 x^{5} - 240 x^{4} y + 720 x^{3} y^{2} - 1080 x^{2} y^{3} + 10 \cdot {(- 3)}^{4} x y^{4} + {(- 3 y)}^{5}$

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

$32 x^{5} - 240 x^{4} y + 720 x^{3} y^{2} - 1080 x^{2} y^{3} + 10 \cdot 81 x y^{4} + {(- 3 y)}^{5}$

Multiply $10$ by $81$ .

$32 x^{5} - 240 x^{4} y + 720 x^{3} y^{2} - 1080 x^{2} y^{3} + 810 x y^{4} + {(- 3 y)}^{5}$

Apply the product rule to $- 3 y$ .

$32 x^{5} - 240 x^{4} y + 720 x^{3} y^{2} - 1080 x^{2} y^{3} + 810 x y^{4} + {(- 3)}^{5} y^{5}$

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

$32 x^{5} - 240 x^{4} y + 720 x^{3} y^{2} - 1080 x^{2} y^{3} + 810 x y^{4} - 243 y^{5}$

Expand the Trigonometric Expression (2x-3y)^5

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