Expand the Trigonometric Expression (2u-v)^6

${(2 u - v)}^{6}$

Step 1

Use the Binomial Theorem.

${(2 u)}^{6} + 6 {(2 u)}^{5} (- v) + 15 {(2 u)}^{4} {(- v)}^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 (2 u) {(- v)}^{5} + {(- v)}^{6}$

Step 2

Simplify each term.

Apply the product rule to $2 u$ .

$2^{6} u^{6} + 6 {(2 u)}^{5} (- v) + 15 {(2 u)}^{4} {(- v)}^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 (2 u) {(- v)}^{5} + {(- v)}^{6}$

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

$64 u^{6} + 6 {(2 u)}^{5} (- v) + 15 {(2 u)}^{4} {(- v)}^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 (2 u) {(- v)}^{5} + {(- v)}^{6}$

Rewrite using the commutative property of multiplication.

$64 u^{6} + 6 \cdot - 1 {(2 u)}^{5} v + 15 {(2 u)}^{4} {(- v)}^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 (2 u) {(- v)}^{5} + {(- v)}^{6}$

Multiply $6$ by $- 1$ .

$64 u^{6} - 6 {(2 u)}^{5} v + 15 {(2 u)}^{4} {(- v)}^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 (2 u) {(- v)}^{5} + {(- v)}^{6}$

Apply the product rule to $2 u$ .

$64 u^{6} - 6 (2^{5} u^{5}) v + 15 {(2 u)}^{4} {(- v)}^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 (2 u) {(- v)}^{5} + {(- v)}^{6}$

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

$64 u^{6} - 6 (32 u^{5}) v + 15 {(2 u)}^{4} {(- v)}^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 (2 u) {(- v)}^{5} + {(- v)}^{6}$

Multiply $32$ by $- 6$ .

$64 u^{6} - 192 u^{5} v + 15 {(2 u)}^{4} {(- v)}^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 (2 u) {(- v)}^{5} + {(- v)}^{6}$

Apply the product rule to $2 u$ .

$64 u^{6} - 192 u^{5} v + 15 (2^{4} u^{4}) {(- v)}^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 (2 u) {(- v)}^{5} + {(- v)}^{6}$

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

$64 u^{6} - 192 u^{5} v + 15 (16 u^{4}) {(- v)}^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 (2 u) {(- v)}^{5} + {(- v)}^{6}$

Multiply $16$ by $15$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} {(- v)}^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 (2 u) {(- v)}^{5} + {(- v)}^{6}$

Apply the product rule to $- v$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} ({(- 1)}^{2} v^{2}) + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 (2 u) {(- v)}^{5} + {(- v)}^{6}$

Rewrite using the commutative property of multiplication.

$64 u^{6} - 192 u^{5} v + 240 \cdot {(- 1)}^{2} u^{4} v^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 (2 u) {(- v)}^{5} + {(- v)}^{6}$

Raise $- 1$ to the power of $2$ .

$64 u^{6} - 192 u^{5} v + 240 \cdot 1 u^{4} v^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 (2 u) {(- v)}^{5} + {(- v)}^{6}$

Multiply $240$ by $1$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 (2 u) {(- v)}^{5} + {(- v)}^{6}$

Apply the product rule to $2 u$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} + 20 (2^{3} u^{3}) {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 (2 u) {(- v)}^{5} + {(- v)}^{6}$

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

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} + 20 (8 u^{3}) {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 (2 u) {(- v)}^{5} + {(- v)}^{6}$

Multiply $8$ by $20$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} + 160 u^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 (2 u) {(- v)}^{5} + {(- v)}^{6}$

Apply the product rule to $- v$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} + 160 u^{3} ({(- 1)}^{3} v^{3}) + 15 {(2 u)}^{2} {(- v)}^{4} + 6 (2 u) {(- v)}^{5} + {(- v)}^{6}$

Rewrite using the commutative property of multiplication.

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} + 160 \cdot {(- 1)}^{3} u^{3} v^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 (2 u) {(- v)}^{5} + {(- v)}^{6}$

Raise $- 1$ to the power of $3$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} + 160 \cdot - 1 u^{3} v^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 (2 u) {(- v)}^{5} + {(- v)}^{6}$

Multiply $160$ by $- 1$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 (2 u) {(- v)}^{5} + {(- v)}^{6}$

Apply the product rule to $2 u$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 15 (2^{2} u^{2}) {(- v)}^{4} + 6 (2 u) {(- v)}^{5} + {(- v)}^{6}$

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

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 15 (4 u^{2}) {(- v)}^{4} + 6 (2 u) {(- v)}^{5} + {(- v)}^{6}$

Multiply $4$ by $15$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 u^{2} {(- v)}^{4} + 6 (2 u) {(- v)}^{5} + {(- v)}^{6}$

Apply the product rule to $- v$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 u^{2} ({(- 1)}^{4} v^{4}) + 6 (2 u) {(- v)}^{5} + {(- v)}^{6}$

Rewrite using the commutative property of multiplication.

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 \cdot {(- 1)}^{4} u^{2} v^{4} + 6 (2 u) {(- v)}^{5} + {(- v)}^{6}$

Raise $- 1$ to the power of $4$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 \cdot 1 u^{2} v^{4} + 6 (2 u) {(- v)}^{5} + {(- v)}^{6}$

Multiply $60$ by $1$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 u^{2} v^{4} + 6 (2 u) {(- v)}^{5} + {(- v)}^{6}$

Multiply $2$ by $6$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 u^{2} v^{4} + 12 u {(- v)}^{5} + {(- v)}^{6}$

Apply the product rule to $- v$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 u^{2} v^{4} + 12 u ({(- 1)}^{5} v^{5}) + {(- v)}^{6}$

Rewrite using the commutative property of multiplication.

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 u^{2} v^{4} + 12 \cdot {(- 1)}^{5} u v^{5} + {(- v)}^{6}$

Raise $- 1$ to the power of $5$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 u^{2} v^{4} + 12 \cdot - 1 u v^{5} + {(- v)}^{6}$

Multiply $12$ by $- 1$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 u^{2} v^{4} - 12 u v^{5} + {(- v)}^{6}$

Apply the product rule to $- v$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 u^{2} v^{4} - 12 u v^{5} + {(- 1)}^{6} v^{6}$

Raise $- 1$ to the power of $6$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 u^{2} v^{4} - 12 u v^{5} + 1 v^{6}$

Multiply $v^{6}$ by $1$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 u^{2} v^{4} - 12 u v^{5} + v^{6}$

Expand the Trigonometric Expression (2u-v)^6

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