Trigonometry Examples

${(x - 2 y)}^{6}$

Step 1

Use the Binomial Theorem.

$x^{6} + 6 x^{5} (- 2 y) + 15 x^{4} {(- 2 y)}^{2} + 20 x^{3} {(- 2 y)}^{3} + 15 x^{2} {(- 2 y)}^{4} + 6 x {(- 2 y)}^{5} + {(- 2 y)}^{6}$

Step 2

Simplify each term.

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Step 2.1

Rewrite using the commutative property of multiplication.

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

Step 2.2

Multiply $6$ by $- 2$ .

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

Step 2.3

Apply the product rule to $- 2 y$ .

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

Step 2.4

Rewrite using the commutative property of multiplication.

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

Step 2.5

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

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

Step 2.6

Multiply $15$ by $4$ .

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

Step 2.7

Apply the product rule to $- 2 y$ .

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

Step 2.8

Rewrite using the commutative property of multiplication.

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

Step 2.9

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

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

Step 2.10

Multiply $20$ by $- 8$ .

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

Step 2.11

Apply the product rule to $- 2 y$ .

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

Step 2.12

Rewrite using the commutative property of multiplication.

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

Step 2.13

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

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

Step 2.14

Multiply $15$ by $16$ .

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

Step 2.15

Apply the product rule to $- 2 y$ .

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

Step 2.16

Rewrite using the commutative property of multiplication.

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

Step 2.17

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

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

Step 2.18

Multiply $6$ by $- 32$ .

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

Step 2.19

Apply the product rule to $- 2 y$ .

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

Step 2.20

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

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