Expand the Trigonometric Expression (n-4)^6

${(n - 4)}^{6}$

Step 1

Use the Binomial Theorem.

$n^{6} + 6 n^{5} \cdot - 4 + 15 n^{4} {(- 4)}^{2} + 20 n^{3} {(- 4)}^{3} + 15 n^{2} {(- 4)}^{4} + 6 n {(- 4)}^{5} + {(- 4)}^{6}$

Step 2

Simplify each term.

Multiply $- 4$ by $6$ .

$n^{6} - 24 n^{5} + 15 n^{4} {(- 4)}^{2} + 20 n^{3} {(- 4)}^{3} + 15 n^{2} {(- 4)}^{4} + 6 n {(- 4)}^{5} + {(- 4)}^{6}$

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

$n^{6} - 24 n^{5} + 15 n^{4} \cdot 16 + 20 n^{3} {(- 4)}^{3} + 15 n^{2} {(- 4)}^{4} + 6 n {(- 4)}^{5} + {(- 4)}^{6}$

Multiply $16$ by $15$ .

$n^{6} - 24 n^{5} + 240 n^{4} + 20 n^{3} {(- 4)}^{3} + 15 n^{2} {(- 4)}^{4} + 6 n {(- 4)}^{5} + {(- 4)}^{6}$

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

$n^{6} - 24 n^{5} + 240 n^{4} + 20 n^{3} \cdot - 64 + 15 n^{2} {(- 4)}^{4} + 6 n {(- 4)}^{5} + {(- 4)}^{6}$

Multiply $- 64$ by $20$ .

$n^{6} - 24 n^{5} + 240 n^{4} - 1280 n^{3} + 15 n^{2} {(- 4)}^{4} + 6 n {(- 4)}^{5} + {(- 4)}^{6}$

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

$n^{6} - 24 n^{5} + 240 n^{4} - 1280 n^{3} + 15 n^{2} \cdot 256 + 6 n {(- 4)}^{5} + {(- 4)}^{6}$

Multiply $256$ by $15$ .

$n^{6} - 24 n^{5} + 240 n^{4} - 1280 n^{3} + 3840 n^{2} + 6 n {(- 4)}^{5} + {(- 4)}^{6}$

Raise $- 4$ to the power of $5$ .

$n^{6} - 24 n^{5} + 240 n^{4} - 1280 n^{3} + 3840 n^{2} + 6 n \cdot - 1024 + {(- 4)}^{6}$

Multiply $- 1024$ by $6$ .

$n^{6} - 24 n^{5} + 240 n^{4} - 1280 n^{3} + 3840 n^{2} - 6144 n + {(- 4)}^{6}$

Raise $- 4$ to the power of $6$ .

$n^{6} - 24 n^{5} + 240 n^{4} - 1280 n^{3} + 3840 n^{2} - 6144 n + 4096$