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

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

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify each term.

Apply the product rule to $3 x$ .

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

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

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

Rewrite using the commutative property of multiplication.

$27 x^{3} + 3 \cdot 4 {(3 x)}^{2} y + 3 (3 x) {(4 y)}^{2} + {(4 y)}^{3}$

Multiply $3$ by $4$ .

$27 x^{3} + 12 {(3 x)}^{2} y + 3 (3 x) {(4 y)}^{2} + {(4 y)}^{3}$

Apply the product rule to $3 x$ .

$27 x^{3} + 12 (3^{2} x^{2}) y + 3 (3 x) {(4 y)}^{2} + {(4 y)}^{3}$

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

$27 x^{3} + 12 (9 x^{2}) y + 3 (3 x) {(4 y)}^{2} + {(4 y)}^{3}$

Multiply $9$ by $12$ .

$27 x^{3} + 108 x^{2} y + 3 (3 x) {(4 y)}^{2} + {(4 y)}^{3}$

Multiply $3$ by $3$ .

$27 x^{3} + 108 x^{2} y + 9 x {(4 y)}^{2} + {(4 y)}^{3}$

Apply the product rule to $4 y$ .

$27 x^{3} + 108 x^{2} y + 9 x (4^{2} y^{2}) + {(4 y)}^{3}$

Rewrite using the commutative property of multiplication.

$27 x^{3} + 108 x^{2} y + 9 \cdot 4^{2} x y^{2} + {(4 y)}^{3}$

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

$27 x^{3} + 108 x^{2} y + 9 \cdot 16 x y^{2} + {(4 y)}^{3}$

Multiply $9$ by $16$ .

$27 x^{3} + 108 x^{2} y + 144 x y^{2} + {(4 y)}^{3}$

Apply the product rule to $4 y$ .

$27 x^{3} + 108 x^{2} y + 144 x y^{2} + 4^{3} y^{3}$

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

$27 x^{3} + 108 x^{2} y + 144 x y^{2} + 64 y^{3}$