Expand the Trigonometric Expression (5x-3y)^3

${(5 x - 3 y)}^{3}$

Step 1

Use the Binomial Theorem.

${(5 x)}^{3} + 3 {(5 x)}^{2} (- 3 y) + 3 (5 x) {(- 3 y)}^{2} + {(- 3 y)}^{3}$

Step 2

Simplify each term.

Apply the product rule to $5 x$ .

$5^{3} x^{3} + 3 {(5 x)}^{2} (- 3 y) + 3 (5 x) {(- 3 y)}^{2} + {(- 3 y)}^{3}$

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

$125 x^{3} + 3 {(5 x)}^{2} (- 3 y) + 3 (5 x) {(- 3 y)}^{2} + {(- 3 y)}^{3}$

Rewrite using the commutative property of multiplication.

$125 x^{3} + 3 \cdot - 3 {(5 x)}^{2} y + 3 (5 x) {(- 3 y)}^{2} + {(- 3 y)}^{3}$

Multiply $3$ by $- 3$ .

$125 x^{3} - 9 {(5 x)}^{2} y + 3 (5 x) {(- 3 y)}^{2} + {(- 3 y)}^{3}$

Apply the product rule to $5 x$ .

$125 x^{3} - 9 (5^{2} x^{2}) y + 3 (5 x) {(- 3 y)}^{2} + {(- 3 y)}^{3}$

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

$125 x^{3} - 9 (25 x^{2}) y + 3 (5 x) {(- 3 y)}^{2} + {(- 3 y)}^{3}$

Multiply $25$ by $- 9$ .

$125 x^{3} - 225 x^{2} y + 3 (5 x) {(- 3 y)}^{2} + {(- 3 y)}^{3}$

Multiply $5$ by $3$ .

$125 x^{3} - 225 x^{2} y + 15 x {(- 3 y)}^{2} + {(- 3 y)}^{3}$

Apply the product rule to $- 3 y$ .

$125 x^{3} - 225 x^{2} y + 15 x ({(- 3)}^{2} y^{2}) + {(- 3 y)}^{3}$

Rewrite using the commutative property of multiplication.

$125 x^{3} - 225 x^{2} y + 15 \cdot {(- 3)}^{2} x y^{2} + {(- 3 y)}^{3}$

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

$125 x^{3} - 225 x^{2} y + 15 \cdot 9 x y^{2} + {(- 3 y)}^{3}$

Multiply $15$ by $9$ .

$125 x^{3} - 225 x^{2} y + 135 x y^{2} + {(- 3 y)}^{3}$

Apply the product rule to $- 3 y$ .

$125 x^{3} - 225 x^{2} y + 135 x y^{2} + {(- 3)}^{3} y^{3}$

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

$125 x^{3} - 225 x^{2} y + 135 x y^{2} - 27 y^{3}$