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

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

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify each term.

Apply the product rule to $2 x$ .

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

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

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

Rewrite using the commutative property of multiplication.

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

Multiply $5$ by $- 2$ .

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

Apply the product rule to $2 x$ .

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

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

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

Multiply $16$ by $- 10$ .

$32 x^{5} - 160 x^{4} y + 10 {(2 x)}^{3} {(- 2 y)}^{2} + 10 {(2 x)}^{2} {(- 2 y)}^{3} + 5 (2 x) {(- 2 y)}^{4} + {(- 2 y)}^{5}$

Apply the product rule to $2 x$ .

$32 x^{5} - 160 x^{4} y + 10 (2^{3} x^{3}) {(- 2 y)}^{2} + 10 {(2 x)}^{2} {(- 2 y)}^{3} + 5 (2 x) {(- 2 y)}^{4} + {(- 2 y)}^{5}$

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

$32 x^{5} - 160 x^{4} y + 10 (8 x^{3}) {(- 2 y)}^{2} + 10 {(2 x)}^{2} {(- 2 y)}^{3} + 5 (2 x) {(- 2 y)}^{4} + {(- 2 y)}^{5}$

Multiply $8$ by $10$ .

$32 x^{5} - 160 x^{4} y + 80 x^{3} {(- 2 y)}^{2} + 10 {(2 x)}^{2} {(- 2 y)}^{3} + 5 (2 x) {(- 2 y)}^{4} + {(- 2 y)}^{5}$

Apply the product rule to $- 2 y$ .

$32 x^{5} - 160 x^{4} y + 80 x^{3} ({(- 2)}^{2} y^{2}) + 10 {(2 x)}^{2} {(- 2 y)}^{3} + 5 (2 x) {(- 2 y)}^{4} + {(- 2 y)}^{5}$

Rewrite using the commutative property of multiplication.

$32 x^{5} - 160 x^{4} y + 80 \cdot {(- 2)}^{2} x^{3} y^{2} + 10 {(2 x)}^{2} {(- 2 y)}^{3} + 5 (2 x) {(- 2 y)}^{4} + {(- 2 y)}^{5}$

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

$32 x^{5} - 160 x^{4} y + 80 \cdot 4 x^{3} y^{2} + 10 {(2 x)}^{2} {(- 2 y)}^{3} + 5 (2 x) {(- 2 y)}^{4} + {(- 2 y)}^{5}$

Multiply $80$ by $4$ .

$32 x^{5} - 160 x^{4} y + 320 x^{3} y^{2} + 10 {(2 x)}^{2} {(- 2 y)}^{3} + 5 (2 x) {(- 2 y)}^{4} + {(- 2 y)}^{5}$

Apply the product rule to $2 x$ .

$32 x^{5} - 160 x^{4} y + 320 x^{3} y^{2} + 10 (2^{2} x^{2}) {(- 2 y)}^{3} + 5 (2 x) {(- 2 y)}^{4} + {(- 2 y)}^{5}$

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

$32 x^{5} - 160 x^{4} y + 320 x^{3} y^{2} + 10 (4 x^{2}) {(- 2 y)}^{3} + 5 (2 x) {(- 2 y)}^{4} + {(- 2 y)}^{5}$

Multiply $4$ by $10$ .

$32 x^{5} - 160 x^{4} y + 320 x^{3} y^{2} + 40 x^{2} {(- 2 y)}^{3} + 5 (2 x) {(- 2 y)}^{4} + {(- 2 y)}^{5}$

Apply the product rule to $- 2 y$ .

$32 x^{5} - 160 x^{4} y + 320 x^{3} y^{2} + 40 x^{2} ({(- 2)}^{3} y^{3}) + 5 (2 x) {(- 2 y)}^{4} + {(- 2 y)}^{5}$

Rewrite using the commutative property of multiplication.

$32 x^{5} - 160 x^{4} y + 320 x^{3} y^{2} + 40 \cdot {(- 2)}^{3} x^{2} y^{3} + 5 (2 x) {(- 2 y)}^{4} + {(- 2 y)}^{5}$

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

$32 x^{5} - 160 x^{4} y + 320 x^{3} y^{2} + 40 \cdot - 8 x^{2} y^{3} + 5 (2 x) {(- 2 y)}^{4} + {(- 2 y)}^{5}$

Multiply $40$ by $- 8$ .

$32 x^{5} - 160 x^{4} y + 320 x^{3} y^{2} - 320 x^{2} y^{3} + 5 (2 x) {(- 2 y)}^{4} + {(- 2 y)}^{5}$

Multiply $2$ by $5$ .

$32 x^{5} - 160 x^{4} y + 320 x^{3} y^{2} - 320 x^{2} y^{3} + 10 x {(- 2 y)}^{4} + {(- 2 y)}^{5}$

Apply the product rule to $- 2 y$ .

$32 x^{5} - 160 x^{4} y + 320 x^{3} y^{2} - 320 x^{2} y^{3} + 10 x ({(- 2)}^{4} y^{4}) + {(- 2 y)}^{5}$

Rewrite using the commutative property of multiplication.

$32 x^{5} - 160 x^{4} y + 320 x^{3} y^{2} - 320 x^{2} y^{3} + 10 \cdot {(- 2)}^{4} x y^{4} + {(- 2 y)}^{5}$

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

$32 x^{5} - 160 x^{4} y + 320 x^{3} y^{2} - 320 x^{2} y^{3} + 10 \cdot 16 x y^{4} + {(- 2 y)}^{5}$

Multiply $10$ by $16$ .

$32 x^{5} - 160 x^{4} y + 320 x^{3} y^{2} - 320 x^{2} y^{3} + 160 x y^{4} + {(- 2 y)}^{5}$

Apply the product rule to $- 2 y$ .

$32 x^{5} - 160 x^{4} y + 320 x^{3} y^{2} - 320 x^{2} y^{3} + 160 x y^{4} + {(- 2)}^{5} y^{5}$

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

$32 x^{5} - 160 x^{4} y + 320 x^{3} y^{2} - 320 x^{2} y^{3} + 160 x y^{4} - 32 y^{5}$

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

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