Expand the Trigonometric Expression (a-2b)^6

${(a - 2 b)}^{6}$

Step 1

Use the Binomial Theorem.

$a^{6} + 6 a^{5} (- 2 b) + 15 a^{4} {(- 2 b)}^{2} + 20 a^{3} {(- 2 b)}^{3} + 15 a^{2} {(- 2 b)}^{4} + 6 a {(- 2 b)}^{5} + {(- 2 b)}^{6}$

Step 2

Simplify each term.

Rewrite using the commutative property of multiplication.

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

Multiply $6$ by $- 2$ .

$a^{6} - 12 a^{5} b + 15 a^{4} {(- 2 b)}^{2} + 20 a^{3} {(- 2 b)}^{3} + 15 a^{2} {(- 2 b)}^{4} + 6 a {(- 2 b)}^{5} + {(- 2 b)}^{6}$

Apply the product rule to $- 2 b$ .

$a^{6} - 12 a^{5} b + 15 a^{4} ({(- 2)}^{2} b^{2}) + 20 a^{3} {(- 2 b)}^{3} + 15 a^{2} {(- 2 b)}^{4} + 6 a {(- 2 b)}^{5} + {(- 2 b)}^{6}$

Rewrite using the commutative property of multiplication.

$a^{6} - 12 a^{5} b + 15 \cdot {(- 2)}^{2} a^{4} b^{2} + 20 a^{3} {(- 2 b)}^{3} + 15 a^{2} {(- 2 b)}^{4} + 6 a {(- 2 b)}^{5} + {(- 2 b)}^{6}$

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

$a^{6} - 12 a^{5} b + 15 \cdot 4 a^{4} b^{2} + 20 a^{3} {(- 2 b)}^{3} + 15 a^{2} {(- 2 b)}^{4} + 6 a {(- 2 b)}^{5} + {(- 2 b)}^{6}$

Multiply $15$ by $4$ .

$a^{6} - 12 a^{5} b + 60 a^{4} b^{2} + 20 a^{3} {(- 2 b)}^{3} + 15 a^{2} {(- 2 b)}^{4} + 6 a {(- 2 b)}^{5} + {(- 2 b)}^{6}$

Apply the product rule to $- 2 b$ .

$a^{6} - 12 a^{5} b + 60 a^{4} b^{2} + 20 a^{3} ({(- 2)}^{3} b^{3}) + 15 a^{2} {(- 2 b)}^{4} + 6 a {(- 2 b)}^{5} + {(- 2 b)}^{6}$

Rewrite using the commutative property of multiplication.

$a^{6} - 12 a^{5} b + 60 a^{4} b^{2} + 20 \cdot {(- 2)}^{3} a^{3} b^{3} + 15 a^{2} {(- 2 b)}^{4} + 6 a {(- 2 b)}^{5} + {(- 2 b)}^{6}$

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

$a^{6} - 12 a^{5} b + 60 a^{4} b^{2} + 20 \cdot - 8 a^{3} b^{3} + 15 a^{2} {(- 2 b)}^{4} + 6 a {(- 2 b)}^{5} + {(- 2 b)}^{6}$

Multiply $20$ by $- 8$ .

$a^{6} - 12 a^{5} b + 60 a^{4} b^{2} - 160 a^{3} b^{3} + 15 a^{2} {(- 2 b)}^{4} + 6 a {(- 2 b)}^{5} + {(- 2 b)}^{6}$

Apply the product rule to $- 2 b$ .

$a^{6} - 12 a^{5} b + 60 a^{4} b^{2} - 160 a^{3} b^{3} + 15 a^{2} ({(- 2)}^{4} b^{4}) + 6 a {(- 2 b)}^{5} + {(- 2 b)}^{6}$

Rewrite using the commutative property of multiplication.

$a^{6} - 12 a^{5} b + 60 a^{4} b^{2} - 160 a^{3} b^{3} + 15 \cdot {(- 2)}^{4} a^{2} b^{4} + 6 a {(- 2 b)}^{5} + {(- 2 b)}^{6}$

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

$a^{6} - 12 a^{5} b + 60 a^{4} b^{2} - 160 a^{3} b^{3} + 15 \cdot 16 a^{2} b^{4} + 6 a {(- 2 b)}^{5} + {(- 2 b)}^{6}$

Multiply $15$ by $16$ .

$a^{6} - 12 a^{5} b + 60 a^{4} b^{2} - 160 a^{3} b^{3} + 240 a^{2} b^{4} + 6 a {(- 2 b)}^{5} + {(- 2 b)}^{6}$

Apply the product rule to $- 2 b$ .

$a^{6} - 12 a^{5} b + 60 a^{4} b^{2} - 160 a^{3} b^{3} + 240 a^{2} b^{4} + 6 a ({(- 2)}^{5} b^{5}) + {(- 2 b)}^{6}$

Rewrite using the commutative property of multiplication.

$a^{6} - 12 a^{5} b + 60 a^{4} b^{2} - 160 a^{3} b^{3} + 240 a^{2} b^{4} + 6 \cdot {(- 2)}^{5} a b^{5} + {(- 2 b)}^{6}$

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

$a^{6} - 12 a^{5} b + 60 a^{4} b^{2} - 160 a^{3} b^{3} + 240 a^{2} b^{4} + 6 \cdot - 32 a b^{5} + {(- 2 b)}^{6}$

Multiply $6$ by $- 32$ .

$a^{6} - 12 a^{5} b + 60 a^{4} b^{2} - 160 a^{3} b^{3} + 240 a^{2} b^{4} - 192 a b^{5} + {(- 2 b)}^{6}$

Apply the product rule to $- 2 b$ .

$a^{6} - 12 a^{5} b + 60 a^{4} b^{2} - 160 a^{3} b^{3} + 240 a^{2} b^{4} - 192 a b^{5} + {(- 2)}^{6} b^{6}$

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

$a^{6} - 12 a^{5} b + 60 a^{4} b^{2} - 160 a^{3} b^{3} + 240 a^{2} b^{4} - 192 a b^{5} + 64 b^{6}$

Expand the Trigonometric Expression (a-2b)^6

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