Expand the Trigonometric Expression (2x-5y)^3

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

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify each term.

Apply the product rule to $2 x$ .

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

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

$8 x^{3} + 3 {(2 x)}^{2} (- 5 y) + 3 (2 x) {(- 5 y)}^{2} + {(- 5 y)}^{3}$

Rewrite using the commutative property of multiplication.

$8 x^{3} + 3 \cdot - 5 {(2 x)}^{2} y + 3 (2 x) {(- 5 y)}^{2} + {(- 5 y)}^{3}$

Multiply $3$ by $- 5$ .

$8 x^{3} - 15 {(2 x)}^{2} y + 3 (2 x) {(- 5 y)}^{2} + {(- 5 y)}^{3}$

Apply the product rule to $2 x$ .

$8 x^{3} - 15 (2^{2} x^{2}) y + 3 (2 x) {(- 5 y)}^{2} + {(- 5 y)}^{3}$

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

$8 x^{3} - 15 (4 x^{2}) y + 3 (2 x) {(- 5 y)}^{2} + {(- 5 y)}^{3}$

Multiply $4$ by $- 15$ .

$8 x^{3} - 60 x^{2} y + 3 (2 x) {(- 5 y)}^{2} + {(- 5 y)}^{3}$

Multiply $2$ by $3$ .

$8 x^{3} - 60 x^{2} y + 6 x {(- 5 y)}^{2} + {(- 5 y)}^{3}$

Apply the product rule to $- 5 y$ .

$8 x^{3} - 60 x^{2} y + 6 x ({(- 5)}^{2} y^{2}) + {(- 5 y)}^{3}$

Rewrite using the commutative property of multiplication.

$8 x^{3} - 60 x^{2} y + 6 \cdot {(- 5)}^{2} x y^{2} + {(- 5 y)}^{3}$

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

$8 x^{3} - 60 x^{2} y + 6 \cdot 25 x y^{2} + {(- 5 y)}^{3}$

Multiply $6$ by $25$ .

$8 x^{3} - 60 x^{2} y + 150 x y^{2} + {(- 5 y)}^{3}$

Apply the product rule to $- 5 y$ .

$8 x^{3} - 60 x^{2} y + 150 x y^{2} + {(- 5)}^{3} y^{3}$

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

$8 x^{3} - 60 x^{2} y + 150 x y^{2} - 125 y^{3}$