Find the Product (3xy-1)^3

${(3 x y - 1)}^{3}$

Step 1

Use the Binomial Theorem.

${(3 x y)}^{3} + 3 {(3 x y)}^{2} \cdot - 1 + 3 (3 x y) {(- 1)}^{2} + {(- 1)}^{3}$

Step 2

Simplify each term.

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $3 x y$ .

${(3 x)}^{3} y^{3} + 3 {(3 x y)}^{2} \cdot - 1 + 3 (3 x y) {(- 1)}^{2} + {(- 1)}^{3}$

Apply the product rule to $3 x$ .

$3^{3} x^{3} y^{3} + 3 {(3 x y)}^{2} \cdot - 1 + 3 (3 x y) {(- 1)}^{2} + {(- 1)}^{3}$

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

$27 x^{3} y^{3} + 3 {(3 x y)}^{2} \cdot - 1 + 3 (3 x y) {(- 1)}^{2} + {(- 1)}^{3}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $3 x y$ .

$27 x^{3} y^{3} + 3 ({(3 x)}^{2} y^{2}) \cdot - 1 + 3 (3 x y) {(- 1)}^{2} + {(- 1)}^{3}$

Apply the product rule to $3 x$ .

$27 x^{3} y^{3} + 3 (3^{2} x^{2} y^{2}) \cdot - 1 + 3 (3 x y) {(- 1)}^{2} + {(- 1)}^{3}$

Multiply $3$ by $3^{2}$ by adding the exponents.

Move $3^{2}$ .

$27 x^{3} y^{3} + 3^{2} \cdot 3 (x^{2} y^{2}) \cdot - 1 + 3 (3 x y) {(- 1)}^{2} + {(- 1)}^{3}$

Multiply $3^{2}$ by $3$ .

Raise $3$ to the power of $1$ .

$27 x^{3} y^{3} + 3^{2} \cdot 3^{1} (x^{2} y^{2}) \cdot - 1 + 3 (3 x y) {(- 1)}^{2} + {(- 1)}^{3}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$27 x^{3} y^{3} + 3^{2 + 1} (x^{2} y^{2}) \cdot - 1 + 3 (3 x y) {(- 1)}^{2} + {(- 1)}^{3}$

Add $2$ and $1$ .

$27 x^{3} y^{3} + 3^{3} (x^{2} y^{2}) \cdot - 1 + 3 (3 x y) {(- 1)}^{2} + {(- 1)}^{3}$

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

$27 x^{3} y^{3} + 27 (x^{2} y^{2}) \cdot - 1 + 3 (3 x y) {(- 1)}^{2} + {(- 1)}^{3}$

Multiply $- 1$ by $27$ .

$27 x^{3} y^{3} - 27 x^{2} y^{2} + 3 (3 x y) {(- 1)}^{2} + {(- 1)}^{3}$

Multiply $3$ by $3$ .

$27 x^{3} y^{3} - 27 x^{2} y^{2} + 9 (x y) {(- 1)}^{2} + {(- 1)}^{3}$

Raise $- 1$ to the power of $2$ .

$27 x^{3} y^{3} - 27 x^{2} y^{2} + 9 x y \cdot 1 + {(- 1)}^{3}$

Multiply $9$ by $1$ .

$27 x^{3} y^{3} - 27 x^{2} y^{2} + 9 x y + {(- 1)}^{3}$

Raise $- 1$ to the power of $3$ .

$27 x^{3} y^{3} - 27 x^{2} y^{2} + 9 x y - 1$