Expand the Trigonometric Expression (x-5y)^8

${(x - 5 y)}^{8}$

Step 1

Use the Binomial Theorem.

$x^{8} + 8 x^{7} (- 5 y) + 28 x^{6} {(- 5 y)}^{2} + 56 x^{5} {(- 5 y)}^{3} + 70 x^{4} {(- 5 y)}^{4} + 56 x^{3} {(- 5 y)}^{5} + 28 x^{2} {(- 5 y)}^{6} + 8 x {(- 5 y)}^{7} + {(- 5 y)}^{8}$

Step 2

Simplify each term.

Rewrite using the commutative property of multiplication.

$x^{8} + 8 \cdot - 5 x^{7} y + 28 x^{6} {(- 5 y)}^{2} + 56 x^{5} {(- 5 y)}^{3} + 70 x^{4} {(- 5 y)}^{4} + 56 x^{3} {(- 5 y)}^{5} + 28 x^{2} {(- 5 y)}^{6} + 8 x {(- 5 y)}^{7} + {(- 5 y)}^{8}$

Multiply $8$ by $- 5$ .

$x^{8} - 40 x^{7} y + 28 x^{6} {(- 5 y)}^{2} + 56 x^{5} {(- 5 y)}^{3} + 70 x^{4} {(- 5 y)}^{4} + 56 x^{3} {(- 5 y)}^{5} + 28 x^{2} {(- 5 y)}^{6} + 8 x {(- 5 y)}^{7} + {(- 5 y)}^{8}$

Apply the product rule to $- 5 y$ .

$x^{8} - 40 x^{7} y + 28 x^{6} ({(- 5)}^{2} y^{2}) + 56 x^{5} {(- 5 y)}^{3} + 70 x^{4} {(- 5 y)}^{4} + 56 x^{3} {(- 5 y)}^{5} + 28 x^{2} {(- 5 y)}^{6} + 8 x {(- 5 y)}^{7} + {(- 5 y)}^{8}$

Rewrite using the commutative property of multiplication.

$x^{8} - 40 x^{7} y + 28 \cdot {(- 5)}^{2} x^{6} y^{2} + 56 x^{5} {(- 5 y)}^{3} + 70 x^{4} {(- 5 y)}^{4} + 56 x^{3} {(- 5 y)}^{5} + 28 x^{2} {(- 5 y)}^{6} + 8 x {(- 5 y)}^{7} + {(- 5 y)}^{8}$

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

$x^{8} - 40 x^{7} y + 28 \cdot 25 x^{6} y^{2} + 56 x^{5} {(- 5 y)}^{3} + 70 x^{4} {(- 5 y)}^{4} + 56 x^{3} {(- 5 y)}^{5} + 28 x^{2} {(- 5 y)}^{6} + 8 x {(- 5 y)}^{7} + {(- 5 y)}^{8}$

Multiply $28$ by $25$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} + 56 x^{5} {(- 5 y)}^{3} + 70 x^{4} {(- 5 y)}^{4} + 56 x^{3} {(- 5 y)}^{5} + 28 x^{2} {(- 5 y)}^{6} + 8 x {(- 5 y)}^{7} + {(- 5 y)}^{8}$

Apply the product rule to $- 5 y$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} + 56 x^{5} ({(- 5)}^{3} y^{3}) + 70 x^{4} {(- 5 y)}^{4} + 56 x^{3} {(- 5 y)}^{5} + 28 x^{2} {(- 5 y)}^{6} + 8 x {(- 5 y)}^{7} + {(- 5 y)}^{8}$

Rewrite using the commutative property of multiplication.

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} + 56 \cdot {(- 5)}^{3} x^{5} y^{3} + 70 x^{4} {(- 5 y)}^{4} + 56 x^{3} {(- 5 y)}^{5} + 28 x^{2} {(- 5 y)}^{6} + 8 x {(- 5 y)}^{7} + {(- 5 y)}^{8}$

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

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} + 56 \cdot - 125 x^{5} y^{3} + 70 x^{4} {(- 5 y)}^{4} + 56 x^{3} {(- 5 y)}^{5} + 28 x^{2} {(- 5 y)}^{6} + 8 x {(- 5 y)}^{7} + {(- 5 y)}^{8}$

Multiply $56$ by $- 125$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 70 x^{4} {(- 5 y)}^{4} + 56 x^{3} {(- 5 y)}^{5} + 28 x^{2} {(- 5 y)}^{6} + 8 x {(- 5 y)}^{7} + {(- 5 y)}^{8}$

Apply the product rule to $- 5 y$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 70 x^{4} ({(- 5)}^{4} y^{4}) + 56 x^{3} {(- 5 y)}^{5} + 28 x^{2} {(- 5 y)}^{6} + 8 x {(- 5 y)}^{7} + {(- 5 y)}^{8}$

Rewrite using the commutative property of multiplication.

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 70 \cdot {(- 5)}^{4} x^{4} y^{4} + 56 x^{3} {(- 5 y)}^{5} + 28 x^{2} {(- 5 y)}^{6} + 8 x {(- 5 y)}^{7} + {(- 5 y)}^{8}$

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

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 70 \cdot 625 x^{4} y^{4} + 56 x^{3} {(- 5 y)}^{5} + 28 x^{2} {(- 5 y)}^{6} + 8 x {(- 5 y)}^{7} + {(- 5 y)}^{8}$

Multiply $70$ by $625$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} + 56 x^{3} {(- 5 y)}^{5} + 28 x^{2} {(- 5 y)}^{6} + 8 x {(- 5 y)}^{7} + {(- 5 y)}^{8}$

Apply the product rule to $- 5 y$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} + 56 x^{3} ({(- 5)}^{5} y^{5}) + 28 x^{2} {(- 5 y)}^{6} + 8 x {(- 5 y)}^{7} + {(- 5 y)}^{8}$

Rewrite using the commutative property of multiplication.

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} + 56 \cdot {(- 5)}^{5} x^{3} y^{5} + 28 x^{2} {(- 5 y)}^{6} + 8 x {(- 5 y)}^{7} + {(- 5 y)}^{8}$

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

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} + 56 \cdot - 3125 x^{3} y^{5} + 28 x^{2} {(- 5 y)}^{6} + 8 x {(- 5 y)}^{7} + {(- 5 y)}^{8}$

Multiply $56$ by $- 3125$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 28 x^{2} {(- 5 y)}^{6} + 8 x {(- 5 y)}^{7} + {(- 5 y)}^{8}$

Apply the product rule to $- 5 y$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 28 x^{2} ({(- 5)}^{6} y^{6}) + 8 x {(- 5 y)}^{7} + {(- 5 y)}^{8}$

Rewrite using the commutative property of multiplication.

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 28 \cdot {(- 5)}^{6} x^{2} y^{6} + 8 x {(- 5 y)}^{7} + {(- 5 y)}^{8}$

Raise $- 5$ to the power of $6$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 28 \cdot 15625 x^{2} y^{6} + 8 x {(- 5 y)}^{7} + {(- 5 y)}^{8}$

Multiply $28$ by $15625$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 437500 x^{2} y^{6} + 8 x {(- 5 y)}^{7} + {(- 5 y)}^{8}$

Apply the product rule to $- 5 y$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 437500 x^{2} y^{6} + 8 x ({(- 5)}^{7} y^{7}) + {(- 5 y)}^{8}$

Rewrite using the commutative property of multiplication.

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 437500 x^{2} y^{6} + 8 \cdot {(- 5)}^{7} x y^{7} + {(- 5 y)}^{8}$

Raise $- 5$ to the power of $7$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 437500 x^{2} y^{6} + 8 \cdot - 78125 x y^{7} + {(- 5 y)}^{8}$

Multiply $8$ by $- 78125$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 437500 x^{2} y^{6} - 625000 x y^{7} + {(- 5 y)}^{8}$

Apply the product rule to $- 5 y$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 437500 x^{2} y^{6} - 625000 x y^{7} + {(- 5)}^{8} y^{8}$

Raise $- 5$ to the power of $8$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 437500 x^{2} y^{6} - 625000 x y^{7} + 390625 y^{8}$

Expand the Trigonometric Expression (x-5y)^8

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