Expand the Trigonometric Expression (3n-2)^5

${(3 n - 2)}^{5}$

Step 1

Use the Binomial Theorem.

${(3 n)}^{5} + 5 {(3 n)}^{4} \cdot - 2 + 10 {(3 n)}^{3} {(- 2)}^{2} + 10 {(3 n)}^{2} {(- 2)}^{3} + 5 (3 n) {(- 2)}^{4} + {(- 2)}^{5}$

Step 2

Simplify each term.

Apply the product rule to $3 n$ .

$3^{5} n^{5} + 5 {(3 n)}^{4} \cdot - 2 + 10 {(3 n)}^{3} {(- 2)}^{2} + 10 {(3 n)}^{2} {(- 2)}^{3} + 5 (3 n) {(- 2)}^{4} + {(- 2)}^{5}$

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

$243 n^{5} + 5 {(3 n)}^{4} \cdot - 2 + 10 {(3 n)}^{3} {(- 2)}^{2} + 10 {(3 n)}^{2} {(- 2)}^{3} + 5 (3 n) {(- 2)}^{4} + {(- 2)}^{5}$

Apply the product rule to $3 n$ .

$243 n^{5} + 5 (3^{4} n^{4}) \cdot - 2 + 10 {(3 n)}^{3} {(- 2)}^{2} + 10 {(3 n)}^{2} {(- 2)}^{3} + 5 (3 n) {(- 2)}^{4} + {(- 2)}^{5}$

Raise $3$ to the power of $4$ .

$243 n^{5} + 5 (81 n^{4}) \cdot - 2 + 10 {(3 n)}^{3} {(- 2)}^{2} + 10 {(3 n)}^{2} {(- 2)}^{3} + 5 (3 n) {(- 2)}^{4} + {(- 2)}^{5}$

Multiply $81$ by $5$ .

$243 n^{5} + 405 n^{4} \cdot - 2 + 10 {(3 n)}^{3} {(- 2)}^{2} + 10 {(3 n)}^{2} {(- 2)}^{3} + 5 (3 n) {(- 2)}^{4} + {(- 2)}^{5}$

Multiply $- 2$ by $405$ .

$243 n^{5} - 810 n^{4} + 10 {(3 n)}^{3} {(- 2)}^{2} + 10 {(3 n)}^{2} {(- 2)}^{3} + 5 (3 n) {(- 2)}^{4} + {(- 2)}^{5}$

Apply the product rule to $3 n$ .

$243 n^{5} - 810 n^{4} + 10 (3^{3} n^{3}) {(- 2)}^{2} + 10 {(3 n)}^{2} {(- 2)}^{3} + 5 (3 n) {(- 2)}^{4} + {(- 2)}^{5}$

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

$243 n^{5} - 810 n^{4} + 10 (27 n^{3}) {(- 2)}^{2} + 10 {(3 n)}^{2} {(- 2)}^{3} + 5 (3 n) {(- 2)}^{4} + {(- 2)}^{5}$

Multiply $27$ by $10$ .

$243 n^{5} - 810 n^{4} + 270 n^{3} {(- 2)}^{2} + 10 {(3 n)}^{2} {(- 2)}^{3} + 5 (3 n) {(- 2)}^{4} + {(- 2)}^{5}$

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

$243 n^{5} - 810 n^{4} + 270 n^{3} \cdot 4 + 10 {(3 n)}^{2} {(- 2)}^{3} + 5 (3 n) {(- 2)}^{4} + {(- 2)}^{5}$

Multiply $4$ by $270$ .

$243 n^{5} - 810 n^{4} + 1080 n^{3} + 10 {(3 n)}^{2} {(- 2)}^{3} + 5 (3 n) {(- 2)}^{4} + {(- 2)}^{5}$

Apply the product rule to $3 n$ .

$243 n^{5} - 810 n^{4} + 1080 n^{3} + 10 (3^{2} n^{2}) {(- 2)}^{3} + 5 (3 n) {(- 2)}^{4} + {(- 2)}^{5}$

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

$243 n^{5} - 810 n^{4} + 1080 n^{3} + 10 (9 n^{2}) {(- 2)}^{3} + 5 (3 n) {(- 2)}^{4} + {(- 2)}^{5}$

Multiply $9$ by $10$ .

$243 n^{5} - 810 n^{4} + 1080 n^{3} + 90 n^{2} {(- 2)}^{3} + 5 (3 n) {(- 2)}^{4} + {(- 2)}^{5}$

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

$243 n^{5} - 810 n^{4} + 1080 n^{3} + 90 n^{2} \cdot - 8 + 5 (3 n) {(- 2)}^{4} + {(- 2)}^{5}$

Multiply $- 8$ by $90$ .

$243 n^{5} - 810 n^{4} + 1080 n^{3} - 720 n^{2} + 5 (3 n) {(- 2)}^{4} + {(- 2)}^{5}$

Multiply $3$ by $5$ .

$243 n^{5} - 810 n^{4} + 1080 n^{3} - 720 n^{2} + 15 n {(- 2)}^{4} + {(- 2)}^{5}$

Raise $- 2$ to the power of $4$ .

$243 n^{5} - 810 n^{4} + 1080 n^{3} - 720 n^{2} + 15 n \cdot 16 + {(- 2)}^{5}$

Multiply $16$ by $15$ .

$243 n^{5} - 810 n^{4} + 1080 n^{3} - 720 n^{2} + 240 n + {(- 2)}^{5}$

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

$243 n^{5} - 810 n^{4} + 1080 n^{3} - 720 n^{2} + 240 n - 32$

Expand the Trigonometric Expression (3n-2)^5

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