Expand the Trigonometric Expression (3a-1)^3

${(3 a - 1)}^{3}$

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify each term.

Apply the product rule to $3 a$ .

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

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

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

Apply the product rule to $3 a$ .

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

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

Move $3^{2}$ .

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

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

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

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

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

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

Add $2$ and $1$ .

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

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

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

Multiply $- 1$ by $27$ .

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

Multiply $3$ by $3$ .

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

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

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

Multiply $9$ by $1$ .

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

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

$27 a^{3} - 27 a^{2} + 9 a - 1$