Expand the Trigonometric Expression (2x-3y)^6

${(2 x - 3 y)}^{6}$

Step 1

Use the Binomial Theorem.

${(2 x)}^{6} + 6 {(2 x)}^{5} (- 3 y) + 15 {(2 x)}^{4} {(- 3 y)}^{2} + 20 {(2 x)}^{3} {(- 3 y)}^{3} + 15 {(2 x)}^{2} {(- 3 y)}^{4} + 6 (2 x) {(- 3 y)}^{5} + {(- 3 y)}^{6}$

Step 2

Simplify each term.

Apply the product rule to $2 x$ .

$2^{6} x^{6} + 6 {(2 x)}^{5} (- 3 y) + 15 {(2 x)}^{4} {(- 3 y)}^{2} + 20 {(2 x)}^{3} {(- 3 y)}^{3} + 15 {(2 x)}^{2} {(- 3 y)}^{4} + 6 (2 x) {(- 3 y)}^{5} + {(- 3 y)}^{6}$

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

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

Rewrite using the commutative property of multiplication.

$64 x^{6} + 6 \cdot - 3 {(2 x)}^{5} y + 15 {(2 x)}^{4} {(- 3 y)}^{2} + 20 {(2 x)}^{3} {(- 3 y)}^{3} + 15 {(2 x)}^{2} {(- 3 y)}^{4} + 6 (2 x) {(- 3 y)}^{5} + {(- 3 y)}^{6}$

Multiply $6$ by $- 3$ .

$64 x^{6} - 18 {(2 x)}^{5} y + 15 {(2 x)}^{4} {(- 3 y)}^{2} + 20 {(2 x)}^{3} {(- 3 y)}^{3} + 15 {(2 x)}^{2} {(- 3 y)}^{4} + 6 (2 x) {(- 3 y)}^{5} + {(- 3 y)}^{6}$

Apply the product rule to $2 x$ .

$64 x^{6} - 18 (2^{5} x^{5}) y + 15 {(2 x)}^{4} {(- 3 y)}^{2} + 20 {(2 x)}^{3} {(- 3 y)}^{3} + 15 {(2 x)}^{2} {(- 3 y)}^{4} + 6 (2 x) {(- 3 y)}^{5} + {(- 3 y)}^{6}$

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

$64 x^{6} - 18 (32 x^{5}) y + 15 {(2 x)}^{4} {(- 3 y)}^{2} + 20 {(2 x)}^{3} {(- 3 y)}^{3} + 15 {(2 x)}^{2} {(- 3 y)}^{4} + 6 (2 x) {(- 3 y)}^{5} + {(- 3 y)}^{6}$

Multiply $32$ by $- 18$ .

$64 x^{6} - 576 x^{5} y + 15 {(2 x)}^{4} {(- 3 y)}^{2} + 20 {(2 x)}^{3} {(- 3 y)}^{3} + 15 {(2 x)}^{2} {(- 3 y)}^{4} + 6 (2 x) {(- 3 y)}^{5} + {(- 3 y)}^{6}$

Apply the product rule to $2 x$ .

$64 x^{6} - 576 x^{5} y + 15 (2^{4} x^{4}) {(- 3 y)}^{2} + 20 {(2 x)}^{3} {(- 3 y)}^{3} + 15 {(2 x)}^{2} {(- 3 y)}^{4} + 6 (2 x) {(- 3 y)}^{5} + {(- 3 y)}^{6}$

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

$64 x^{6} - 576 x^{5} y + 15 (16 x^{4}) {(- 3 y)}^{2} + 20 {(2 x)}^{3} {(- 3 y)}^{3} + 15 {(2 x)}^{2} {(- 3 y)}^{4} + 6 (2 x) {(- 3 y)}^{5} + {(- 3 y)}^{6}$

Multiply $16$ by $15$ .

$64 x^{6} - 576 x^{5} y + 240 x^{4} {(- 3 y)}^{2} + 20 {(2 x)}^{3} {(- 3 y)}^{3} + 15 {(2 x)}^{2} {(- 3 y)}^{4} + 6 (2 x) {(- 3 y)}^{5} + {(- 3 y)}^{6}$

Apply the product rule to $- 3 y$ .

$64 x^{6} - 576 x^{5} y + 240 x^{4} ({(- 3)}^{2} y^{2}) + 20 {(2 x)}^{3} {(- 3 y)}^{3} + 15 {(2 x)}^{2} {(- 3 y)}^{4} + 6 (2 x) {(- 3 y)}^{5} + {(- 3 y)}^{6}$

Rewrite using the commutative property of multiplication.

$64 x^{6} - 576 x^{5} y + 240 \cdot {(- 3)}^{2} x^{4} y^{2} + 20 {(2 x)}^{3} {(- 3 y)}^{3} + 15 {(2 x)}^{2} {(- 3 y)}^{4} + 6 (2 x) {(- 3 y)}^{5} + {(- 3 y)}^{6}$

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

$64 x^{6} - 576 x^{5} y + 240 \cdot 9 x^{4} y^{2} + 20 {(2 x)}^{3} {(- 3 y)}^{3} + 15 {(2 x)}^{2} {(- 3 y)}^{4} + 6 (2 x) {(- 3 y)}^{5} + {(- 3 y)}^{6}$

Multiply $240$ by $9$ .

$64 x^{6} - 576 x^{5} y + 2160 x^{4} y^{2} + 20 {(2 x)}^{3} {(- 3 y)}^{3} + 15 {(2 x)}^{2} {(- 3 y)}^{4} + 6 (2 x) {(- 3 y)}^{5} + {(- 3 y)}^{6}$

Apply the product rule to $2 x$ .

$64 x^{6} - 576 x^{5} y + 2160 x^{4} y^{2} + 20 (2^{3} x^{3}) {(- 3 y)}^{3} + 15 {(2 x)}^{2} {(- 3 y)}^{4} + 6 (2 x) {(- 3 y)}^{5} + {(- 3 y)}^{6}$

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

$64 x^{6} - 576 x^{5} y + 2160 x^{4} y^{2} + 20 (8 x^{3}) {(- 3 y)}^{3} + 15 {(2 x)}^{2} {(- 3 y)}^{4} + 6 (2 x) {(- 3 y)}^{5} + {(- 3 y)}^{6}$

Multiply $8$ by $20$ .

$64 x^{6} - 576 x^{5} y + 2160 x^{4} y^{2} + 160 x^{3} {(- 3 y)}^{3} + 15 {(2 x)}^{2} {(- 3 y)}^{4} + 6 (2 x) {(- 3 y)}^{5} + {(- 3 y)}^{6}$

Apply the product rule to $- 3 y$ .

$64 x^{6} - 576 x^{5} y + 2160 x^{4} y^{2} + 160 x^{3} ({(- 3)}^{3} y^{3}) + 15 {(2 x)}^{2} {(- 3 y)}^{4} + 6 (2 x) {(- 3 y)}^{5} + {(- 3 y)}^{6}$

Rewrite using the commutative property of multiplication.

$64 x^{6} - 576 x^{5} y + 2160 x^{4} y^{2} + 160 \cdot {(- 3)}^{3} x^{3} y^{3} + 15 {(2 x)}^{2} {(- 3 y)}^{4} + 6 (2 x) {(- 3 y)}^{5} + {(- 3 y)}^{6}$

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

$64 x^{6} - 576 x^{5} y + 2160 x^{4} y^{2} + 160 \cdot - 27 x^{3} y^{3} + 15 {(2 x)}^{2} {(- 3 y)}^{4} + 6 (2 x) {(- 3 y)}^{5} + {(- 3 y)}^{6}$

Multiply $160$ by $- 27$ .

$64 x^{6} - 576 x^{5} y + 2160 x^{4} y^{2} - 4320 x^{3} y^{3} + 15 {(2 x)}^{2} {(- 3 y)}^{4} + 6 (2 x) {(- 3 y)}^{5} + {(- 3 y)}^{6}$

Apply the product rule to $2 x$ .

$64 x^{6} - 576 x^{5} y + 2160 x^{4} y^{2} - 4320 x^{3} y^{3} + 15 (2^{2} x^{2}) {(- 3 y)}^{4} + 6 (2 x) {(- 3 y)}^{5} + {(- 3 y)}^{6}$

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

$64 x^{6} - 576 x^{5} y + 2160 x^{4} y^{2} - 4320 x^{3} y^{3} + 15 (4 x^{2}) {(- 3 y)}^{4} + 6 (2 x) {(- 3 y)}^{5} + {(- 3 y)}^{6}$

Multiply $4$ by $15$ .

$64 x^{6} - 576 x^{5} y + 2160 x^{4} y^{2} - 4320 x^{3} y^{3} + 60 x^{2} {(- 3 y)}^{4} + 6 (2 x) {(- 3 y)}^{5} + {(- 3 y)}^{6}$

Apply the product rule to $- 3 y$ .

$64 x^{6} - 576 x^{5} y + 2160 x^{4} y^{2} - 4320 x^{3} y^{3} + 60 x^{2} ({(- 3)}^{4} y^{4}) + 6 (2 x) {(- 3 y)}^{5} + {(- 3 y)}^{6}$

Rewrite using the commutative property of multiplication.

$64 x^{6} - 576 x^{5} y + 2160 x^{4} y^{2} - 4320 x^{3} y^{3} + 60 \cdot {(- 3)}^{4} x^{2} y^{4} + 6 (2 x) {(- 3 y)}^{5} + {(- 3 y)}^{6}$

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

$64 x^{6} - 576 x^{5} y + 2160 x^{4} y^{2} - 4320 x^{3} y^{3} + 60 \cdot 81 x^{2} y^{4} + 6 (2 x) {(- 3 y)}^{5} + {(- 3 y)}^{6}$

Multiply $60$ by $81$ .

$64 x^{6} - 576 x^{5} y + 2160 x^{4} y^{2} - 4320 x^{3} y^{3} + 4860 x^{2} y^{4} + 6 (2 x) {(- 3 y)}^{5} + {(- 3 y)}^{6}$

Multiply $2$ by $6$ .

$64 x^{6} - 576 x^{5} y + 2160 x^{4} y^{2} - 4320 x^{3} y^{3} + 4860 x^{2} y^{4} + 12 x {(- 3 y)}^{5} + {(- 3 y)}^{6}$

Apply the product rule to $- 3 y$ .

$64 x^{6} - 576 x^{5} y + 2160 x^{4} y^{2} - 4320 x^{3} y^{3} + 4860 x^{2} y^{4} + 12 x ({(- 3)}^{5} y^{5}) + {(- 3 y)}^{6}$

Rewrite using the commutative property of multiplication.

$64 x^{6} - 576 x^{5} y + 2160 x^{4} y^{2} - 4320 x^{3} y^{3} + 4860 x^{2} y^{4} + 12 \cdot {(- 3)}^{5} x y^{5} + {(- 3 y)}^{6}$

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

$64 x^{6} - 576 x^{5} y + 2160 x^{4} y^{2} - 4320 x^{3} y^{3} + 4860 x^{2} y^{4} + 12 \cdot - 243 x y^{5} + {(- 3 y)}^{6}$

Multiply $12$ by $- 243$ .

$64 x^{6} - 576 x^{5} y + 2160 x^{4} y^{2} - 4320 x^{3} y^{3} + 4860 x^{2} y^{4} - 2916 x y^{5} + {(- 3 y)}^{6}$

Apply the product rule to $- 3 y$ .

$64 x^{6} - 576 x^{5} y + 2160 x^{4} y^{2} - 4320 x^{3} y^{3} + 4860 x^{2} y^{4} - 2916 x y^{5} + {(- 3)}^{6} y^{6}$

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

$64 x^{6} - 576 x^{5} y + 2160 x^{4} y^{2} - 4320 x^{3} y^{3} + 4860 x^{2} y^{4} - 2916 x y^{5} + 729 y^{6}$

Expand the Trigonometric Expression (2x-3y)^6

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