Simplify (3ab-2de)^3

${(3 a b - 2 d e)}^{3}$

Use the Binomial Theorem.

${(3 a b)}^{3} + 3 {(3 a b)}^{2} (- 2 d e) + 3 (3 a b) {(- 2 d e)}^{2} + {(- 2 d e)}^{3}$

Simplify each term.

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

Apply the product rule to $3 a b$ .

${(3 a)}^{3} b^{3} + 3 {(3 a b)}^{2} (- 2 d e) + 3 (3 a b) {(- 2 d e)}^{2} + {(- 2 d e)}^{3}$

Apply the product rule to $3 a$ .

$3^{3} a^{3} b^{3} + 3 {(3 a b)}^{2} (- 2 d e) + 3 (3 a b) {(- 2 d e)}^{2} + {(- 2 d e)}^{3}$

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

$27 a^{3} b^{3} + 3 {(3 a b)}^{2} (- 2 d e) + 3 (3 a b) {(- 2 d e)}^{2} + {(- 2 d e)}^{3}$

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

Apply the product rule to $3 a b$ .

$27 a^{3} b^{3} + 3 ({(3 a)}^{2} b^{2}) (- 2 d e) + 3 (3 a b) {(- 2 d e)}^{2} + {(- 2 d e)}^{3}$

Apply the product rule to $3 a$ .

$27 a^{3} b^{3} + 3 (3^{2} a^{2} b^{2}) (- 2 d e) + 3 (3 a b) {(- 2 d e)}^{2} + {(- 2 d e)}^{3}$

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

Move $3^{2}$ .

$27 a^{3} b^{3} + 3^{2} \cdot 3 (a^{2} b^{2}) (- 2 d e) + 3 (3 a b) {(- 2 d e)}^{2} + {(- 2 d e)}^{3}$

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

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

$27 a^{3} b^{3} + 3^{2} \cdot 3^{1} (a^{2} b^{2}) (- 2 d e) + 3 (3 a b) {(- 2 d e)}^{2} + {(- 2 d e)}^{3}$

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

$27 a^{3} b^{3} + 3^{2 + 1} (a^{2} b^{2}) (- 2 d e) + 3 (3 a b) {(- 2 d e)}^{2} + {(- 2 d e)}^{3}$

Add $2$ and $1$ .

$27 a^{3} b^{3} + 3^{3} (a^{2} b^{2}) (- 2 d e) + 3 (3 a b) {(- 2 d e)}^{2} + {(- 2 d e)}^{3}$

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

$27 a^{3} b^{3} + 27 (a^{2} b^{2}) (- 2 d e) + 3 (3 a b) {(- 2 d e)}^{2} + {(- 2 d e)}^{3}$

Multiply $- 2$ by $27$ .

$27 a^{3} b^{3} - 54 a^{2} b^{2} (d e) + 3 (3 a b) {(- 2 d e)}^{2} + {(- 2 d e)}^{3}$

Multiply $3$ by $3$ .

$27 a^{3} b^{3} - 54 a^{2} b^{2} d e + 9 (a b) {(- 2 d e)}^{2} + {(- 2 d e)}^{3}$

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

Apply the product rule to $- 2 d e$ .

$27 a^{3} b^{3} - 54 a^{2} b^{2} d e + 9 a b ({(- 2 d)}^{2} e^{2}) + {(- 2 d e)}^{3}$

Apply the product rule to $- 2 d$ .

$27 a^{3} b^{3} - 54 a^{2} b^{2} d e + 9 a b ({(- 2)}^{2} d^{2} e^{2}) + {(- 2 d e)}^{3}$

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

$27 a^{3} b^{3} - 54 a^{2} b^{2} d e + 9 a b (4 d^{2} e^{2}) + {(- 2 d e)}^{3}$

Multiply $4$ by $9$ .

$27 a^{3} b^{3} - 54 a^{2} b^{2} d e + 36 a b (d^{2} e^{2}) + {(- 2 d e)}^{3}$

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

Apply the product rule to $- 2 d e$ .

$27 a^{3} b^{3} - 54 a^{2} b^{2} d e + 36 a b d^{2} e^{2} + {(- 2 d)}^{3} e^{3}$

Apply the product rule to $- 2 d$ .

$27 a^{3} b^{3} - 54 a^{2} b^{2} d e + 36 a b d^{2} e^{2} + {(- 2)}^{3} d^{3} e^{3}$

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

$27 a^{3} b^{3} - 54 a^{2} b^{2} d e + 36 a b d^{2} e^{2} - 8 d^{3} e^{3}$

Simplify (3ab-2de)^3

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