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

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

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify each term.

Apply the product rule to $2 x$ .

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

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

$16 x^{4} + 4 {(2 x)}^{3} (3 y) + 6 {(2 x)}^{2} {(3 y)}^{2} + 4 (2 x) {(3 y)}^{3} + {(3 y)}^{4}$

Rewrite using the commutative property of multiplication.

$16 x^{4} + 4 \cdot 3 {(2 x)}^{3} y + 6 {(2 x)}^{2} {(3 y)}^{2} + 4 (2 x) {(3 y)}^{3} + {(3 y)}^{4}$

Multiply $4$ by $3$ .

$16 x^{4} + 12 {(2 x)}^{3} y + 6 {(2 x)}^{2} {(3 y)}^{2} + 4 (2 x) {(3 y)}^{3} + {(3 y)}^{4}$

Apply the product rule to $2 x$ .

$16 x^{4} + 12 (2^{3} x^{3}) y + 6 {(2 x)}^{2} {(3 y)}^{2} + 4 (2 x) {(3 y)}^{3} + {(3 y)}^{4}$

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

$16 x^{4} + 12 (8 x^{3}) y + 6 {(2 x)}^{2} {(3 y)}^{2} + 4 (2 x) {(3 y)}^{3} + {(3 y)}^{4}$

Multiply $8$ by $12$ .

$16 x^{4} + 96 x^{3} y + 6 {(2 x)}^{2} {(3 y)}^{2} + 4 (2 x) {(3 y)}^{3} + {(3 y)}^{4}$

Apply the product rule to $2 x$ .

$16 x^{4} + 96 x^{3} y + 6 (2^{2} x^{2}) {(3 y)}^{2} + 4 (2 x) {(3 y)}^{3} + {(3 y)}^{4}$

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

$16 x^{4} + 96 x^{3} y + 6 (4 x^{2}) {(3 y)}^{2} + 4 (2 x) {(3 y)}^{3} + {(3 y)}^{4}$

Multiply $4$ by $6$ .

$16 x^{4} + 96 x^{3} y + 24 x^{2} {(3 y)}^{2} + 4 (2 x) {(3 y)}^{3} + {(3 y)}^{4}$

Apply the product rule to $3 y$ .

$16 x^{4} + 96 x^{3} y + 24 x^{2} (3^{2} y^{2}) + 4 (2 x) {(3 y)}^{3} + {(3 y)}^{4}$

Rewrite using the commutative property of multiplication.

$16 x^{4} + 96 x^{3} y + 24 \cdot 3^{2} x^{2} y^{2} + 4 (2 x) {(3 y)}^{3} + {(3 y)}^{4}$

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

$16 x^{4} + 96 x^{3} y + 24 \cdot 9 x^{2} y^{2} + 4 (2 x) {(3 y)}^{3} + {(3 y)}^{4}$

Multiply $24$ by $9$ .

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

Multiply $2$ by $4$ .

$16 x^{4} + 96 x^{3} y + 216 x^{2} y^{2} + 8 x {(3 y)}^{3} + {(3 y)}^{4}$

Apply the product rule to $3 y$ .

$16 x^{4} + 96 x^{3} y + 216 x^{2} y^{2} + 8 x (3^{3} y^{3}) + {(3 y)}^{4}$

Rewrite using the commutative property of multiplication.

$16 x^{4} + 96 x^{3} y + 216 x^{2} y^{2} + 8 \cdot 3^{3} x y^{3} + {(3 y)}^{4}$

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

$16 x^{4} + 96 x^{3} y + 216 x^{2} y^{2} + 8 \cdot 27 x y^{3} + {(3 y)}^{4}$

Multiply $8$ by $27$ .

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

Apply the product rule to $3 y$ .

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

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

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

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

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