Expand the Trigonometric Expression (3u-v)^6

${(3 u - v)}^{6}$

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify each term.

Apply the product rule to $3 u$ .

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

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

$729 u^{6} + 6 {(3 u)}^{5} (- v) + 15 {(3 u)}^{4} {(- v)}^{2} + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 (3 u) {(- v)}^{5} + {(- v)}^{6}$

Rewrite using the commutative property of multiplication.

$729 u^{6} + 6 \cdot - 1 {(3 u)}^{5} v + 15 {(3 u)}^{4} {(- v)}^{2} + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 (3 u) {(- v)}^{5} + {(- v)}^{6}$

Multiply $6$ by $- 1$ .

$729 u^{6} - 6 {(3 u)}^{5} v + 15 {(3 u)}^{4} {(- v)}^{2} + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 (3 u) {(- v)}^{5} + {(- v)}^{6}$

Apply the product rule to $3 u$ .

$729 u^{6} - 6 (3^{5} u^{5}) v + 15 {(3 u)}^{4} {(- v)}^{2} + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 (3 u) {(- v)}^{5} + {(- v)}^{6}$

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

$729 u^{6} - 6 (243 u^{5}) v + 15 {(3 u)}^{4} {(- v)}^{2} + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 (3 u) {(- v)}^{5} + {(- v)}^{6}$

Multiply $243$ by $- 6$ .

$729 u^{6} - 1458 u^{5} v + 15 {(3 u)}^{4} {(- v)}^{2} + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 (3 u) {(- v)}^{5} + {(- v)}^{6}$

Apply the product rule to $3 u$ .

$729 u^{6} - 1458 u^{5} v + 15 (3^{4} u^{4}) {(- v)}^{2} + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 (3 u) {(- v)}^{5} + {(- v)}^{6}$

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

$729 u^{6} - 1458 u^{5} v + 15 (81 u^{4}) {(- v)}^{2} + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 (3 u) {(- v)}^{5} + {(- v)}^{6}$

Multiply $81$ by $15$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} {(- v)}^{2} + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 (3 u) {(- v)}^{5} + {(- v)}^{6}$

Apply the product rule to $- v$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} ({(- 1)}^{2} v^{2}) + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 (3 u) {(- v)}^{5} + {(- v)}^{6}$

Rewrite using the commutative property of multiplication.

$729 u^{6} - 1458 u^{5} v + 1215 \cdot {(- 1)}^{2} u^{4} v^{2} + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 (3 u) {(- v)}^{5} + {(- v)}^{6}$

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

$729 u^{6} - 1458 u^{5} v + 1215 \cdot 1 u^{4} v^{2} + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 (3 u) {(- v)}^{5} + {(- v)}^{6}$

Multiply $1215$ by $1$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 (3 u) {(- v)}^{5} + {(- v)}^{6}$

Apply the product rule to $3 u$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} + 20 (3^{3} u^{3}) {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 (3 u) {(- v)}^{5} + {(- v)}^{6}$

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

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} + 20 (27 u^{3}) {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 (3 u) {(- v)}^{5} + {(- v)}^{6}$

Multiply $27$ by $20$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} + 540 u^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 (3 u) {(- v)}^{5} + {(- v)}^{6}$

Apply the product rule to $- v$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} + 540 u^{3} ({(- 1)}^{3} v^{3}) + 15 {(3 u)}^{2} {(- v)}^{4} + 6 (3 u) {(- v)}^{5} + {(- v)}^{6}$

Rewrite using the commutative property of multiplication.

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} + 540 \cdot {(- 1)}^{3} u^{3} v^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 (3 u) {(- v)}^{5} + {(- v)}^{6}$

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

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} + 540 \cdot - 1 u^{3} v^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 (3 u) {(- v)}^{5} + {(- v)}^{6}$

Multiply $540$ by $- 1$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 (3 u) {(- v)}^{5} + {(- v)}^{6}$

Apply the product rule to $3 u$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 15 (3^{2} u^{2}) {(- v)}^{4} + 6 (3 u) {(- v)}^{5} + {(- v)}^{6}$

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

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 15 (9 u^{2}) {(- v)}^{4} + 6 (3 u) {(- v)}^{5} + {(- v)}^{6}$

Multiply $9$ by $15$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 u^{2} {(- v)}^{4} + 6 (3 u) {(- v)}^{5} + {(- v)}^{6}$

Apply the product rule to $- v$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 u^{2} ({(- 1)}^{4} v^{4}) + 6 (3 u) {(- v)}^{5} + {(- v)}^{6}$

Rewrite using the commutative property of multiplication.

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 \cdot {(- 1)}^{4} u^{2} v^{4} + 6 (3 u) {(- v)}^{5} + {(- v)}^{6}$

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

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 \cdot 1 u^{2} v^{4} + 6 (3 u) {(- v)}^{5} + {(- v)}^{6}$

Multiply $135$ by $1$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 u^{2} v^{4} + 6 (3 u) {(- v)}^{5} + {(- v)}^{6}$

Multiply $3$ by $6$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 u^{2} v^{4} + 18 u {(- v)}^{5} + {(- v)}^{6}$

Apply the product rule to $- v$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 u^{2} v^{4} + 18 u ({(- 1)}^{5} v^{5}) + {(- v)}^{6}$

Rewrite using the commutative property of multiplication.

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 u^{2} v^{4} + 18 \cdot {(- 1)}^{5} u v^{5} + {(- v)}^{6}$

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

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 u^{2} v^{4} + 18 \cdot - 1 u v^{5} + {(- v)}^{6}$

Multiply $18$ by $- 1$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 u^{2} v^{4} - 18 u v^{5} + {(- v)}^{6}$

Apply the product rule to $- v$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 u^{2} v^{4} - 18 u v^{5} + {(- 1)}^{6} v^{6}$

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

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 u^{2} v^{4} - 18 u v^{5} + 1 v^{6}$

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

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 u^{2} v^{4} - 18 u v^{5} + v^{6}$

Expand the Trigonometric Expression (3u-v)^6

Related Questions

Support LCMGCF.com by disabling your adblocker.