Expand the Trigonometric Expression (2-9b)^7

${(2 - 9 b)}^{7}$

Step 1

Use the Binomial Theorem.

$2^{7} + 7 \cdot 2^{6} (- 9 b) + 21 \cdot 2^{5} {(- 9 b)}^{2} + 35 \cdot 2^{4} {(- 9 b)}^{3} + 35 \cdot 2^{3} {(- 9 b)}^{4} + 21 \cdot 2^{2} {(- 9 b)}^{5} + 7 \cdot 2 {(- 9 b)}^{6} + {(- 9 b)}^{7}$

Step 2

Simplify each term.

Raise $2$ to the power of $7$ .

$128 + 7 \cdot 2^{6} (- 9 b) + 21 \cdot 2^{5} {(- 9 b)}^{2} + 35 \cdot 2^{4} {(- 9 b)}^{3} + 35 \cdot 2^{3} {(- 9 b)}^{4} + 21 \cdot 2^{2} {(- 9 b)}^{5} + 7 \cdot 2 {(- 9 b)}^{6} + {(- 9 b)}^{7}$

Raise $2$ to the power of $6$ .

$128 + 7 \cdot 64 (- 9 b) + 21 \cdot 2^{5} {(- 9 b)}^{2} + 35 \cdot 2^{4} {(- 9 b)}^{3} + 35 \cdot 2^{3} {(- 9 b)}^{4} + 21 \cdot 2^{2} {(- 9 b)}^{5} + 7 \cdot 2 {(- 9 b)}^{6} + {(- 9 b)}^{7}$

Multiply $7$ by $64$ .

$128 + 448 (- 9 b) + 21 \cdot 2^{5} {(- 9 b)}^{2} + 35 \cdot 2^{4} {(- 9 b)}^{3} + 35 \cdot 2^{3} {(- 9 b)}^{4} + 21 \cdot 2^{2} {(- 9 b)}^{5} + 7 \cdot 2 {(- 9 b)}^{6} + {(- 9 b)}^{7}$

Multiply $- 9$ by $448$ .

$128 - 4032 b + 21 \cdot 2^{5} {(- 9 b)}^{2} + 35 \cdot 2^{4} {(- 9 b)}^{3} + 35 \cdot 2^{3} {(- 9 b)}^{4} + 21 \cdot 2^{2} {(- 9 b)}^{5} + 7 \cdot 2 {(- 9 b)}^{6} + {(- 9 b)}^{7}$

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

$128 - 4032 b + 21 \cdot 32 {(- 9 b)}^{2} + 35 \cdot 2^{4} {(- 9 b)}^{3} + 35 \cdot 2^{3} {(- 9 b)}^{4} + 21 \cdot 2^{2} {(- 9 b)}^{5} + 7 \cdot 2 {(- 9 b)}^{6} + {(- 9 b)}^{7}$

Multiply $21$ by $32$ .

$128 - 4032 b + 672 {(- 9 b)}^{2} + 35 \cdot 2^{4} {(- 9 b)}^{3} + 35 \cdot 2^{3} {(- 9 b)}^{4} + 21 \cdot 2^{2} {(- 9 b)}^{5} + 7 \cdot 2 {(- 9 b)}^{6} + {(- 9 b)}^{7}$

Apply the product rule to $- 9 b$ .

$128 - 4032 b + 672 ({(- 9)}^{2} b^{2}) + 35 \cdot 2^{4} {(- 9 b)}^{3} + 35 \cdot 2^{3} {(- 9 b)}^{4} + 21 \cdot 2^{2} {(- 9 b)}^{5} + 7 \cdot 2 {(- 9 b)}^{6} + {(- 9 b)}^{7}$

Raise $- 9$ to the power of $2$ .

$128 - 4032 b + 672 (81 b^{2}) + 35 \cdot 2^{4} {(- 9 b)}^{3} + 35 \cdot 2^{3} {(- 9 b)}^{4} + 21 \cdot 2^{2} {(- 9 b)}^{5} + 7 \cdot 2 {(- 9 b)}^{6} + {(- 9 b)}^{7}$

Multiply $81$ by $672$ .

$128 - 4032 b + 54432 b^{2} + 35 \cdot 2^{4} {(- 9 b)}^{3} + 35 \cdot 2^{3} {(- 9 b)}^{4} + 21 \cdot 2^{2} {(- 9 b)}^{5} + 7 \cdot 2 {(- 9 b)}^{6} + {(- 9 b)}^{7}$

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

$128 - 4032 b + 54432 b^{2} + 35 \cdot 16 {(- 9 b)}^{3} + 35 \cdot 2^{3} {(- 9 b)}^{4} + 21 \cdot 2^{2} {(- 9 b)}^{5} + 7 \cdot 2 {(- 9 b)}^{6} + {(- 9 b)}^{7}$

Multiply $35$ by $16$ .

$128 - 4032 b + 54432 b^{2} + 560 {(- 9 b)}^{3} + 35 \cdot 2^{3} {(- 9 b)}^{4} + 21 \cdot 2^{2} {(- 9 b)}^{5} + 7 \cdot 2 {(- 9 b)}^{6} + {(- 9 b)}^{7}$

Apply the product rule to $- 9 b$ .

$128 - 4032 b + 54432 b^{2} + 560 ({(- 9)}^{3} b^{3}) + 35 \cdot 2^{3} {(- 9 b)}^{4} + 21 \cdot 2^{2} {(- 9 b)}^{5} + 7 \cdot 2 {(- 9 b)}^{6} + {(- 9 b)}^{7}$

Raise $- 9$ to the power of $3$ .

$128 - 4032 b + 54432 b^{2} + 560 (- 729 b^{3}) + 35 \cdot 2^{3} {(- 9 b)}^{4} + 21 \cdot 2^{2} {(- 9 b)}^{5} + 7 \cdot 2 {(- 9 b)}^{6} + {(- 9 b)}^{7}$

Multiply $- 729$ by $560$ .

$128 - 4032 b + 54432 b^{2} - 408240 b^{3} + 35 \cdot 2^{3} {(- 9 b)}^{4} + 21 \cdot 2^{2} {(- 9 b)}^{5} + 7 \cdot 2 {(- 9 b)}^{6} + {(- 9 b)}^{7}$

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

$128 - 4032 b + 54432 b^{2} - 408240 b^{3} + 35 \cdot 8 {(- 9 b)}^{4} + 21 \cdot 2^{2} {(- 9 b)}^{5} + 7 \cdot 2 {(- 9 b)}^{6} + {(- 9 b)}^{7}$

Multiply $35$ by $8$ .

$128 - 4032 b + 54432 b^{2} - 408240 b^{3} + 280 {(- 9 b)}^{4} + 21 \cdot 2^{2} {(- 9 b)}^{5} + 7 \cdot 2 {(- 9 b)}^{6} + {(- 9 b)}^{7}$

Apply the product rule to $- 9 b$ .

$128 - 4032 b + 54432 b^{2} - 408240 b^{3} + 280 ({(- 9)}^{4} b^{4}) + 21 \cdot 2^{2} {(- 9 b)}^{5} + 7 \cdot 2 {(- 9 b)}^{6} + {(- 9 b)}^{7}$

Raise $- 9$ to the power of $4$ .

$128 - 4032 b + 54432 b^{2} - 408240 b^{3} + 280 (6561 b^{4}) + 21 \cdot 2^{2} {(- 9 b)}^{5} + 7 \cdot 2 {(- 9 b)}^{6} + {(- 9 b)}^{7}$

Multiply $6561$ by $280$ .

$128 - 4032 b + 54432 b^{2} - 408240 b^{3} + 1837080 b^{4} + 21 \cdot 2^{2} {(- 9 b)}^{5} + 7 \cdot 2 {(- 9 b)}^{6} + {(- 9 b)}^{7}$

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

$128 - 4032 b + 54432 b^{2} - 408240 b^{3} + 1837080 b^{4} + 21 \cdot 4 {(- 9 b)}^{5} + 7 \cdot 2 {(- 9 b)}^{6} + {(- 9 b)}^{7}$

Multiply $21$ by $4$ .

$128 - 4032 b + 54432 b^{2} - 408240 b^{3} + 1837080 b^{4} + 84 {(- 9 b)}^{5} + 7 \cdot 2 {(- 9 b)}^{6} + {(- 9 b)}^{7}$

Apply the product rule to $- 9 b$ .

$128 - 4032 b + 54432 b^{2} - 408240 b^{3} + 1837080 b^{4} + 84 ({(- 9)}^{5} b^{5}) + 7 \cdot 2 {(- 9 b)}^{6} + {(- 9 b)}^{7}$

Raise $- 9$ to the power of $5$ .

$128 - 4032 b + 54432 b^{2} - 408240 b^{3} + 1837080 b^{4} + 84 (- 59049 b^{5}) + 7 \cdot 2 {(- 9 b)}^{6} + {(- 9 b)}^{7}$

Multiply $- 59049$ by $84$ .

$128 - 4032 b + 54432 b^{2} - 408240 b^{3} + 1837080 b^{4} - 4960116 b^{5} + 7 \cdot 2 {(- 9 b)}^{6} + {(- 9 b)}^{7}$

Multiply $7$ by $2$ .

$128 - 4032 b + 54432 b^{2} - 408240 b^{3} + 1837080 b^{4} - 4960116 b^{5} + 14 {(- 9 b)}^{6} + {(- 9 b)}^{7}$

Apply the product rule to $- 9 b$ .

$128 - 4032 b + 54432 b^{2} - 408240 b^{3} + 1837080 b^{4} - 4960116 b^{5} + 14 ({(- 9)}^{6} b^{6}) + {(- 9 b)}^{7}$

Raise $- 9$ to the power of $6$ .

$128 - 4032 b + 54432 b^{2} - 408240 b^{3} + 1837080 b^{4} - 4960116 b^{5} + 14 (531441 b^{6}) + {(- 9 b)}^{7}$

Multiply $531441$ by $14$ .

$128 - 4032 b + 54432 b^{2} - 408240 b^{3} + 1837080 b^{4} - 4960116 b^{5} + 7440174 b^{6} + {(- 9 b)}^{7}$

Apply the product rule to $- 9 b$ .

$128 - 4032 b + 54432 b^{2} - 408240 b^{3} + 1837080 b^{4} - 4960116 b^{5} + 7440174 b^{6} + {(- 9)}^{7} b^{7}$

Raise $- 9$ to the power of $7$ .

$128 - 4032 b + 54432 b^{2} - 408240 b^{3} + 1837080 b^{4} - 4960116 b^{5} + 7440174 b^{6} - 4782969 b^{7}$

Expand the Trigonometric Expression (2-9b)^7

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