Expand the Trigonometric Expression (3x+2y)^4

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

Use the Binomial Theorem.

${(3 x)}^{4} + 4 {(3 x)}^{3} (2 y) + 6 {(3 x)}^{2} {(2 y)}^{2} + 4 (3 x) {(2 y)}^{3} + {(2 y)}^{4}$

Simplify each term.

Apply the product rule to $3 x$ .

$3^{4} x^{4} + 4 {(3 x)}^{3} (2 y) + 6 {(3 x)}^{2} {(2 y)}^{2} + 4 (3 x) {(2 y)}^{3} + {(2 y)}^{4}$

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

$81 x^{4} + 4 {(3 x)}^{3} (2 y) + 6 {(3 x)}^{2} {(2 y)}^{2} + 4 (3 x) {(2 y)}^{3} + {(2 y)}^{4}$

Rewrite using the commutative property of multiplication.

$81 x^{4} + 4 \cdot 2 {(3 x)}^{3} y + 6 {(3 x)}^{2} {(2 y)}^{2} + 4 (3 x) {(2 y)}^{3} + {(2 y)}^{4}$

Multiply $4$ by $2$ .

$81 x^{4} + 8 {(3 x)}^{3} y + 6 {(3 x)}^{2} {(2 y)}^{2} + 4 (3 x) {(2 y)}^{3} + {(2 y)}^{4}$

Apply the product rule to $3 x$ .

$81 x^{4} + 8 (3^{3} x^{3}) y + 6 {(3 x)}^{2} {(2 y)}^{2} + 4 (3 x) {(2 y)}^{3} + {(2 y)}^{4}$

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

$81 x^{4} + 8 (27 x^{3}) y + 6 {(3 x)}^{2} {(2 y)}^{2} + 4 (3 x) {(2 y)}^{3} + {(2 y)}^{4}$

Multiply $27$ by $8$ .

$81 x^{4} + 216 x^{3} y + 6 {(3 x)}^{2} {(2 y)}^{2} + 4 (3 x) {(2 y)}^{3} + {(2 y)}^{4}$

Apply the product rule to $3 x$ .

$81 x^{4} + 216 x^{3} y + 6 (3^{2} x^{2}) {(2 y)}^{2} + 4 (3 x) {(2 y)}^{3} + {(2 y)}^{4}$

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

$81 x^{4} + 216 x^{3} y + 6 (9 x^{2}) {(2 y)}^{2} + 4 (3 x) {(2 y)}^{3} + {(2 y)}^{4}$

Multiply $9$ by $6$ .

$81 x^{4} + 216 x^{3} y + 54 x^{2} {(2 y)}^{2} + 4 (3 x) {(2 y)}^{3} + {(2 y)}^{4}$

Apply the product rule to $2 y$ .

$81 x^{4} + 216 x^{3} y + 54 x^{2} (2^{2} y^{2}) + 4 (3 x) {(2 y)}^{3} + {(2 y)}^{4}$

Rewrite using the commutative property of multiplication.

$81 x^{4} + 216 x^{3} y + 54 \cdot 2^{2} x^{2} y^{2} + 4 (3 x) {(2 y)}^{3} + {(2 y)}^{4}$

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

$81 x^{4} + 216 x^{3} y + 54 \cdot 4 x^{2} y^{2} + 4 (3 x) {(2 y)}^{3} + {(2 y)}^{4}$

Multiply $54$ by $4$ .

$81 x^{4} + 216 x^{3} y + 216 x^{2} y^{2} + 4 (3 x) {(2 y)}^{3} + {(2 y)}^{4}$

Multiply $3$ by $4$ .

$81 x^{4} + 216 x^{3} y + 216 x^{2} y^{2} + 12 x {(2 y)}^{3} + {(2 y)}^{4}$

Apply the product rule to $2 y$ .

$81 x^{4} + 216 x^{3} y + 216 x^{2} y^{2} + 12 x (2^{3} y^{3}) + {(2 y)}^{4}$

Rewrite using the commutative property of multiplication.

$81 x^{4} + 216 x^{3} y + 216 x^{2} y^{2} + 12 \cdot 2^{3} x y^{3} + {(2 y)}^{4}$

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

$81 x^{4} + 216 x^{3} y + 216 x^{2} y^{2} + 12 \cdot 8 x y^{3} + {(2 y)}^{4}$

Multiply $12$ by $8$ .

$81 x^{4} + 216 x^{3} y + 216 x^{2} y^{2} + 96 x y^{3} + {(2 y)}^{4}$

Apply the product rule to $2 y$ .

$81 x^{4} + 216 x^{3} y + 216 x^{2} y^{2} + 96 x y^{3} + 2^{4} y^{4}$

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

$81 x^{4} + 216 x^{3} y + 216 x^{2} y^{2} + 96 x y^{3} + 16 y^{4}$

Expand the Trigonometric Expression (3x+2y)^4

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