Expand the Trigonometric Expression (4x-5y^2)^2(4x+5y^2)^2

${(4 x - 5 y^{2})}^{2} {(4 x + 5 y^{2})}^{2}$

Step 1

Rewrite ${(4 x - 5 y^{2})}^{2}$ as $(4 x - 5 y^{2}) (4 x - 5 y^{2})$ .

$(4 x - 5 y^{2}) (4 x - 5 y^{2}) {(4 x + 5 y^{2})}^{2}$

Step 2

Expand $(4 x - 5 y^{2}) (4 x - 5 y^{2})$ using the FOIL Method.

Apply the distributive property.

$(4 x (4 x - 5 y^{2}) - 5 y^{2} (4 x - 5 y^{2})) {(4 x + 5 y^{2})}^{2}$

Apply the distributive property.

$(4 x (4 x) + 4 x (- 5 y^{2}) - 5 y^{2} (4 x - 5 y^{2})) {(4 x + 5 y^{2})}^{2}$

Apply the distributive property.

$(4 x (4 x) + 4 x (- 5 y^{2}) - 5 y^{2} (4 x) - 5 y^{2} (- 5 y^{2})) {(4 x + 5 y^{2})}^{2}$

Step 3

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$(4 \cdot 4 x \cdot x + 4 x (- 5 y^{2}) - 5 y^{2} (4 x) - 5 y^{2} (- 5 y^{2})) {(4 x + 5 y^{2})}^{2}$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$(4 \cdot 4 (x \cdot x) + 4 x (- 5 y^{2}) - 5 y^{2} (4 x) - 5 y^{2} (- 5 y^{2})) {(4 x + 5 y^{2})}^{2}$

Multiply $x$ by $x$ .

$(4 \cdot 4 x^{2} + 4 x (- 5 y^{2}) - 5 y^{2} (4 x) - 5 y^{2} (- 5 y^{2})) {(4 x + 5 y^{2})}^{2}$

Multiply $4$ by $4$ .

$(16 x^{2} + 4 x (- 5 y^{2}) - 5 y^{2} (4 x) - 5 y^{2} (- 5 y^{2})) {(4 x + 5 y^{2})}^{2}$

Rewrite using the commutative property of multiplication.

$(16 x^{2} + 4 \cdot - 5 x y^{2} - 5 y^{2} (4 x) - 5 y^{2} (- 5 y^{2})) {(4 x + 5 y^{2})}^{2}$

Multiply $4$ by $- 5$ .

$(16 x^{2} - 20 x y^{2} - 5 y^{2} (4 x) - 5 y^{2} (- 5 y^{2})) {(4 x + 5 y^{2})}^{2}$

Rewrite using the commutative property of multiplication.

$(16 x^{2} - 20 x y^{2} - 5 \cdot 4 y^{2} x - 5 y^{2} (- 5 y^{2})) {(4 x + 5 y^{2})}^{2}$

Multiply $- 5$ by $4$ .

$(16 x^{2} - 20 x y^{2} - 20 y^{2} x - 5 y^{2} (- 5 y^{2})) {(4 x + 5 y^{2})}^{2}$

Rewrite using the commutative property of multiplication.

$(16 x^{2} - 20 x y^{2} - 20 y^{2} x - 5 \cdot - 5 y^{2} y^{2}) {(4 x + 5 y^{2})}^{2}$

Multiply $y^{2}$ by $y^{2}$ by adding the exponents.

Move $y^{2}$ .

$(16 x^{2} - 20 x y^{2} - 20 y^{2} x - 5 \cdot - 5 (y^{2} y^{2})) {(4 x + 5 y^{2})}^{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(16 x^{2} - 20 x y^{2} - 20 y^{2} x - 5 \cdot - 5 y^{2 + 2}) {(4 x + 5 y^{2})}^{2}$

Add $2$ and $2$ .

$(16 x^{2} - 20 x y^{2} - 20 y^{2} x - 5 \cdot - 5 y^{4}) {(4 x + 5 y^{2})}^{2}$

Multiply $- 5$ by $- 5$ .

$(16 x^{2} - 20 x y^{2} - 20 y^{2} x + 25 y^{4}) {(4 x + 5 y^{2})}^{2}$

Subtract $20 y^{2} x$ from $- 20 x y^{2}$ .

Move $y^{2}$ .

$(16 x^{2} - 20 x y^{2} - 20 x y^{2} + 25 y^{4}) {(4 x + 5 y^{2})}^{2}$

Subtract $20 x y^{2}$ from $- 20 x y^{2}$ .

$(16 x^{2} - 40 x y^{2} + 25 y^{4}) {(4 x + 5 y^{2})}^{2}$

Step 4

Rewrite ${(4 x + 5 y^{2})}^{2}$ as $(4 x + 5 y^{2}) (4 x + 5 y^{2})$ .

$(16 x^{2} - 40 x y^{2} + 25 y^{4}) ((4 x + 5 y^{2}) (4 x + 5 y^{2}))$

Step 5

Expand $(4 x + 5 y^{2}) (4 x + 5 y^{2})$ using the FOIL Method.

Apply the distributive property.

$(16 x^{2} - 40 x y^{2} + 25 y^{4}) (4 x (4 x + 5 y^{2}) + 5 y^{2} (4 x + 5 y^{2}))$

Apply the distributive property.

$(16 x^{2} - 40 x y^{2} + 25 y^{4}) (4 x (4 x) + 4 x (5 y^{2}) + 5 y^{2} (4 x + 5 y^{2}))$

Apply the distributive property.

$(16 x^{2} - 40 x y^{2} + 25 y^{4}) (4 x (4 x) + 4 x (5 y^{2}) + 5 y^{2} (4 x) + 5 y^{2} (5 y^{2}))$

Step 6

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$(16 x^{2} - 40 x y^{2} + 25 y^{4}) (4 \cdot 4 x \cdot x + 4 x (5 y^{2}) + 5 y^{2} (4 x) + 5 y^{2} (5 y^{2}))$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$(16 x^{2} - 40 x y^{2} + 25 y^{4}) (4 \cdot 4 (x \cdot x) + 4 x (5 y^{2}) + 5 y^{2} (4 x) + 5 y^{2} (5 y^{2}))$

Multiply $x$ by $x$ .

$(16 x^{2} - 40 x y^{2} + 25 y^{4}) (4 \cdot 4 x^{2} + 4 x (5 y^{2}) + 5 y^{2} (4 x) + 5 y^{2} (5 y^{2}))$

Multiply $4$ by $4$ .

$(16 x^{2} - 40 x y^{2} + 25 y^{4}) (16 x^{2} + 4 x (5 y^{2}) + 5 y^{2} (4 x) + 5 y^{2} (5 y^{2}))$

Rewrite using the commutative property of multiplication.

$(16 x^{2} - 40 x y^{2} + 25 y^{4}) (16 x^{2} + 4 \cdot 5 x y^{2} + 5 y^{2} (4 x) + 5 y^{2} (5 y^{2}))$

Multiply $4$ by $5$ .

$(16 x^{2} - 40 x y^{2} + 25 y^{4}) (16 x^{2} + 20 x y^{2} + 5 y^{2} (4 x) + 5 y^{2} (5 y^{2}))$

Rewrite using the commutative property of multiplication.

$(16 x^{2} - 40 x y^{2} + 25 y^{4}) (16 x^{2} + 20 x y^{2} + 5 \cdot 4 y^{2} x + 5 y^{2} (5 y^{2}))$

Multiply $5$ by $4$ .

$(16 x^{2} - 40 x y^{2} + 25 y^{4}) (16 x^{2} + 20 x y^{2} + 20 y^{2} x + 5 y^{2} (5 y^{2}))$

Rewrite using the commutative property of multiplication.

$(16 x^{2} - 40 x y^{2} + 25 y^{4}) (16 x^{2} + 20 x y^{2} + 20 y^{2} x + 5 \cdot 5 y^{2} y^{2})$

Multiply $y^{2}$ by $y^{2}$ by adding the exponents.

Move $y^{2}$ .

$(16 x^{2} - 40 x y^{2} + 25 y^{4}) (16 x^{2} + 20 x y^{2} + 20 y^{2} x + 5 \cdot 5 (y^{2} y^{2}))$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(16 x^{2} - 40 x y^{2} + 25 y^{4}) (16 x^{2} + 20 x y^{2} + 20 y^{2} x + 5 \cdot 5 y^{2 + 2})$

Add $2$ and $2$ .

$(16 x^{2} - 40 x y^{2} + 25 y^{4}) (16 x^{2} + 20 x y^{2} + 20 y^{2} x + 5 \cdot 5 y^{4})$

Multiply $5$ by $5$ .

$(16 x^{2} - 40 x y^{2} + 25 y^{4}) (16 x^{2} + 20 x y^{2} + 20 y^{2} x + 25 y^{4})$

Add $20 x y^{2}$ and $20 y^{2} x$ .

Move $y^{2}$ .

$(16 x^{2} - 40 x y^{2} + 25 y^{4}) (16 x^{2} + 20 x y^{2} + 20 x y^{2} + 25 y^{4})$

Add $20 x y^{2}$ and $20 x y^{2}$ .

$(16 x^{2} - 40 x y^{2} + 25 y^{4}) (16 x^{2} + 40 x y^{2} + 25 y^{4})$

Step 7

Expand $(16 x^{2} - 40 x y^{2} + 25 y^{4}) (16 x^{2} + 40 x y^{2} + 25 y^{4})$ by multiplying each term in the first expression by each term in the second expression.

$16 x^{2} (16 x^{2}) + 16 x^{2} (40 x y^{2}) + 16 x^{2} (25 y^{4}) - 40 x y^{2} (16 x^{2}) - 40 x y^{2} (40 x y^{2}) - 40 x y^{2} (25 y^{4}) + 25 y^{4} (16 x^{2}) + 25 y^{4} (40 x y^{2}) + 25 y^{4} (25 y^{4})$

Step 8

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$16 \cdot 16 x^{2} x^{2} + 16 x^{2} (40 x y^{2}) + 16 x^{2} (25 y^{4}) - 40 x y^{2} (16 x^{2}) - 40 x y^{2} (40 x y^{2}) - 40 x y^{2} (25 y^{4}) + 25 y^{4} (16 x^{2}) + 25 y^{4} (40 x y^{2}) + 25 y^{4} (25 y^{4})$

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$16 \cdot 16 (x^{2} x^{2}) + 16 x^{2} (40 x y^{2}) + 16 x^{2} (25 y^{4}) - 40 x y^{2} (16 x^{2}) - 40 x y^{2} (40 x y^{2}) - 40 x y^{2} (25 y^{4}) + 25 y^{4} (16 x^{2}) + 25 y^{4} (40 x y^{2}) + 25 y^{4} (25 y^{4})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$16 \cdot 16 x^{2 + 2} + 16 x^{2} (40 x y^{2}) + 16 x^{2} (25 y^{4}) - 40 x y^{2} (16 x^{2}) - 40 x y^{2} (40 x y^{2}) - 40 x y^{2} (25 y^{4}) + 25 y^{4} (16 x^{2}) + 25 y^{4} (40 x y^{2}) + 25 y^{4} (25 y^{4})$

Add $2$ and $2$ .

$16 \cdot 16 x^{4} + 16 x^{2} (40 x y^{2}) + 16 x^{2} (25 y^{4}) - 40 x y^{2} (16 x^{2}) - 40 x y^{2} (40 x y^{2}) - 40 x y^{2} (25 y^{4}) + 25 y^{4} (16 x^{2}) + 25 y^{4} (40 x y^{2}) + 25 y^{4} (25 y^{4})$

Multiply $16$ by $16$ .

$256 x^{4} + 16 x^{2} (40 x y^{2}) + 16 x^{2} (25 y^{4}) - 40 x y^{2} (16 x^{2}) - 40 x y^{2} (40 x y^{2}) - 40 x y^{2} (25 y^{4}) + 25 y^{4} (16 x^{2}) + 25 y^{4} (40 x y^{2}) + 25 y^{4} (25 y^{4})$

Multiply $x^{2}$ by $x$ by adding the exponents.

Move $x$ .

$256 x^{4} + 16 (x \cdot x^{2}) (40 y^{2}) + 16 x^{2} (25 y^{4}) - 40 x y^{2} (16 x^{2}) - 40 x y^{2} (40 x y^{2}) - 40 x y^{2} (25 y^{4}) + 25 y^{4} (16 x^{2}) + 25 y^{4} (40 x y^{2}) + 25 y^{4} (25 y^{4})$

Multiply $x$ by $x^{2}$ .

Raise $x$ to the power of $1$ .

$256 x^{4} + 16 (x^{1} x^{2}) (40 y^{2}) + 16 x^{2} (25 y^{4}) - 40 x y^{2} (16 x^{2}) - 40 x y^{2} (40 x y^{2}) - 40 x y^{2} (25 y^{4}) + 25 y^{4} (16 x^{2}) + 25 y^{4} (40 x y^{2}) + 25 y^{4} (25 y^{4})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$256 x^{4} + 16 x^{1 + 2} (40 y^{2}) + 16 x^{2} (25 y^{4}) - 40 x y^{2} (16 x^{2}) - 40 x y^{2} (40 x y^{2}) - 40 x y^{2} (25 y^{4}) + 25 y^{4} (16 x^{2}) + 25 y^{4} (40 x y^{2}) + 25 y^{4} (25 y^{4})$

Add $1$ and $2$ .

$256 x^{4} + 16 x^{3} (40 y^{2}) + 16 x^{2} (25 y^{4}) - 40 x y^{2} (16 x^{2}) - 40 x y^{2} (40 x y^{2}) - 40 x y^{2} (25 y^{4}) + 25 y^{4} (16 x^{2}) + 25 y^{4} (40 x y^{2}) + 25 y^{4} (25 y^{4})$

Rewrite using the commutative property of multiplication.

$256 x^{4} + 16 \cdot 40 x^{3} y^{2} + 16 x^{2} (25 y^{4}) - 40 x y^{2} (16 x^{2}) - 40 x y^{2} (40 x y^{2}) - 40 x y^{2} (25 y^{4}) + 25 y^{4} (16 x^{2}) + 25 y^{4} (40 x y^{2}) + 25 y^{4} (25 y^{4})$

Multiply $16$ by $40$ .

$256 x^{4} + 640 x^{3} y^{2} + 16 x^{2} (25 y^{4}) - 40 x y^{2} (16 x^{2}) - 40 x y^{2} (40 x y^{2}) - 40 x y^{2} (25 y^{4}) + 25 y^{4} (16 x^{2}) + 25 y^{4} (40 x y^{2}) + 25 y^{4} (25 y^{4})$

Rewrite using the commutative property of multiplication.

$256 x^{4} + 640 x^{3} y^{2} + 16 \cdot 25 x^{2} y^{4} - 40 x y^{2} (16 x^{2}) - 40 x y^{2} (40 x y^{2}) - 40 x y^{2} (25 y^{4}) + 25 y^{4} (16 x^{2}) + 25 y^{4} (40 x y^{2}) + 25 y^{4} (25 y^{4})$

Multiply $16$ by $25$ .

$256 x^{4} + 640 x^{3} y^{2} + 400 x^{2} y^{4} - 40 x y^{2} (16 x^{2}) - 40 x y^{2} (40 x y^{2}) - 40 x y^{2} (25 y^{4}) + 25 y^{4} (16 x^{2}) + 25 y^{4} (40 x y^{2}) + 25 y^{4} (25 y^{4})$

Multiply $x$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$256 x^{4} + 640 x^{3} y^{2} + 400 x^{2} y^{4} - 40 (x^{2} x) y^{2} \cdot 16 - 40 x y^{2} (40 x y^{2}) - 40 x y^{2} (25 y^{4}) + 25 y^{4} (16 x^{2}) + 25 y^{4} (40 x y^{2}) + 25 y^{4} (25 y^{4})$

Multiply $x^{2}$ by $x$ .

Raise $x$ to the power of $1$ .

$256 x^{4} + 640 x^{3} y^{2} + 400 x^{2} y^{4} - 40 (x^{2} x^{1}) y^{2} \cdot 16 - 40 x y^{2} (40 x y^{2}) - 40 x y^{2} (25 y^{4}) + 25 y^{4} (16 x^{2}) + 25 y^{4} (40 x y^{2}) + 25 y^{4} (25 y^{4})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$256 x^{4} + 640 x^{3} y^{2} + 400 x^{2} y^{4} - 40 x^{2 + 1} y^{2} \cdot 16 - 40 x y^{2} (40 x y^{2}) - 40 x y^{2} (25 y^{4}) + 25 y^{4} (16 x^{2}) + 25 y^{4} (40 x y^{2}) + 25 y^{4} (25 y^{4})$

Add $2$ and $1$ .

$256 x^{4} + 640 x^{3} y^{2} + 400 x^{2} y^{4} - 40 x^{3} y^{2} \cdot 16 - 40 x y^{2} (40 x y^{2}) - 40 x y^{2} (25 y^{4}) + 25 y^{4} (16 x^{2}) + 25 y^{4} (40 x y^{2}) + 25 y^{4} (25 y^{4})$

Multiply $16$ by $- 40$ .

$256 x^{4} + 640 x^{3} y^{2} + 400 x^{2} y^{4} - 640 x^{3} y^{2} - 40 x y^{2} (40 x y^{2}) - 40 x y^{2} (25 y^{4}) + 25 y^{4} (16 x^{2}) + 25 y^{4} (40 x y^{2}) + 25 y^{4} (25 y^{4})$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$256 x^{4} + 640 x^{3} y^{2} + 400 x^{2} y^{4} - 640 x^{3} y^{2} - 40 (x \cdot x) y^{2} (40 y^{2}) - 40 x y^{2} (25 y^{4}) + 25 y^{4} (16 x^{2}) + 25 y^{4} (40 x y^{2}) + 25 y^{4} (25 y^{4})$

Multiply $x$ by $x$ .

$256 x^{4} + 640 x^{3} y^{2} + 400 x^{2} y^{4} - 640 x^{3} y^{2} - 40 x^{2} y^{2} (40 y^{2}) - 40 x y^{2} (25 y^{4}) + 25 y^{4} (16 x^{2}) + 25 y^{4} (40 x y^{2}) + 25 y^{4} (25 y^{4})$

Multiply $y^{2}$ by $y^{2}$ by adding the exponents.

Move $y^{2}$ .

$256 x^{4} + 640 x^{3} y^{2} + 400 x^{2} y^{4} - 640 x^{3} y^{2} - 40 x^{2} (y^{2} y^{2}) \cdot 40 - 40 x y^{2} (25 y^{4}) + 25 y^{4} (16 x^{2}) + 25 y^{4} (40 x y^{2}) + 25 y^{4} (25 y^{4})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$256 x^{4} + 640 x^{3} y^{2} + 400 x^{2} y^{4} - 640 x^{3} y^{2} - 40 x^{2} y^{2 + 2} \cdot 40 - 40 x y^{2} (25 y^{4}) + 25 y^{4} (16 x^{2}) + 25 y^{4} (40 x y^{2}) + 25 y^{4} (25 y^{4})$

Add $2$ and $2$ .

$256 x^{4} + 640 x^{3} y^{2} + 400 x^{2} y^{4} - 640 x^{3} y^{2} - 40 x^{2} y^{4} \cdot 40 - 40 x y^{2} (25 y^{4}) + 25 y^{4} (16 x^{2}) + 25 y^{4} (40 x y^{2}) + 25 y^{4} (25 y^{4})$

Multiply $40$ by $- 40$ .

$256 x^{4} + 640 x^{3} y^{2} + 400 x^{2} y^{4} - 640 x^{3} y^{2} - 1600 x^{2} y^{4} - 40 x y^{2} (25 y^{4}) + 25 y^{4} (16 x^{2}) + 25 y^{4} (40 x y^{2}) + 25 y^{4} (25 y^{4})$

Multiply $y^{2}$ by $y^{4}$ by adding the exponents.

Move $y^{4}$ .

$256 x^{4} + 640 x^{3} y^{2} + 400 x^{2} y^{4} - 640 x^{3} y^{2} - 1600 x^{2} y^{4} - 40 x (y^{4} y^{2}) \cdot 25 + 25 y^{4} (16 x^{2}) + 25 y^{4} (40 x y^{2}) + 25 y^{4} (25 y^{4})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$256 x^{4} + 640 x^{3} y^{2} + 400 x^{2} y^{4} - 640 x^{3} y^{2} - 1600 x^{2} y^{4} - 40 x y^{4 + 2} \cdot 25 + 25 y^{4} (16 x^{2}) + 25 y^{4} (40 x y^{2}) + 25 y^{4} (25 y^{4})$

Add $4$ and $2$ .

$256 x^{4} + 640 x^{3} y^{2} + 400 x^{2} y^{4} - 640 x^{3} y^{2} - 1600 x^{2} y^{4} - 40 x y^{6} \cdot 25 + 25 y^{4} (16 x^{2}) + 25 y^{4} (40 x y^{2}) + 25 y^{4} (25 y^{4})$

Multiply $25$ by $- 40$ .

$256 x^{4} + 640 x^{3} y^{2} + 400 x^{2} y^{4} - 640 x^{3} y^{2} - 1600 x^{2} y^{4} - 1000 x y^{6} + 25 y^{4} (16 x^{2}) + 25 y^{4} (40 x y^{2}) + 25 y^{4} (25 y^{4})$

Rewrite using the commutative property of multiplication.

$256 x^{4} + 640 x^{3} y^{2} + 400 x^{2} y^{4} - 640 x^{3} y^{2} - 1600 x^{2} y^{4} - 1000 x y^{6} + 25 \cdot 16 y^{4} x^{2} + 25 y^{4} (40 x y^{2}) + 25 y^{4} (25 y^{4})$

Multiply $25$ by $16$ .

$256 x^{4} + 640 x^{3} y^{2} + 400 x^{2} y^{4} - 640 x^{3} y^{2} - 1600 x^{2} y^{4} - 1000 x y^{6} + 400 y^{4} x^{2} + 25 y^{4} (40 x y^{2}) + 25 y^{4} (25 y^{4})$

Multiply $y^{4}$ by $y^{2}$ by adding the exponents.

Move $y^{2}$ .

$256 x^{4} + 640 x^{3} y^{2} + 400 x^{2} y^{4} - 640 x^{3} y^{2} - 1600 x^{2} y^{4} - 1000 x y^{6} + 400 y^{4} x^{2} + 25 (y^{2} y^{4}) (40 x) + 25 y^{4} (25 y^{4})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$256 x^{4} + 640 x^{3} y^{2} + 400 x^{2} y^{4} - 640 x^{3} y^{2} - 1600 x^{2} y^{4} - 1000 x y^{6} + 400 y^{4} x^{2} + 25 y^{2 + 4} (40 x) + 25 y^{4} (25 y^{4})$

Add $2$ and $4$ .

$256 x^{4} + 640 x^{3} y^{2} + 400 x^{2} y^{4} - 640 x^{3} y^{2} - 1600 x^{2} y^{4} - 1000 x y^{6} + 400 y^{4} x^{2} + 25 y^{6} (40 x) + 25 y^{4} (25 y^{4})$

Rewrite using the commutative property of multiplication.

$256 x^{4} + 640 x^{3} y^{2} + 400 x^{2} y^{4} - 640 x^{3} y^{2} - 1600 x^{2} y^{4} - 1000 x y^{6} + 400 y^{4} x^{2} + 25 \cdot 40 y^{6} x + 25 y^{4} (25 y^{4})$

Multiply $25$ by $40$ .

$256 x^{4} + 640 x^{3} y^{2} + 400 x^{2} y^{4} - 640 x^{3} y^{2} - 1600 x^{2} y^{4} - 1000 x y^{6} + 400 y^{4} x^{2} + 1000 y^{6} x + 25 y^{4} (25 y^{4})$

Rewrite using the commutative property of multiplication.

$256 x^{4} + 640 x^{3} y^{2} + 400 x^{2} y^{4} - 640 x^{3} y^{2} - 1600 x^{2} y^{4} - 1000 x y^{6} + 400 y^{4} x^{2} + 1000 y^{6} x + 25 \cdot 25 y^{4} y^{4}$

Multiply $y^{4}$ by $y^{4}$ by adding the exponents.

Move $y^{4}$ .

$256 x^{4} + 640 x^{3} y^{2} + 400 x^{2} y^{4} - 640 x^{3} y^{2} - 1600 x^{2} y^{4} - 1000 x y^{6} + 400 y^{4} x^{2} + 1000 y^{6} x + 25 \cdot 25 (y^{4} y^{4})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$256 x^{4} + 640 x^{3} y^{2} + 400 x^{2} y^{4} - 640 x^{3} y^{2} - 1600 x^{2} y^{4} - 1000 x y^{6} + 400 y^{4} x^{2} + 1000 y^{6} x + 25 \cdot 25 y^{4 + 4}$

Add $4$ and $4$ .

$256 x^{4} + 640 x^{3} y^{2} + 400 x^{2} y^{4} - 640 x^{3} y^{2} - 1600 x^{2} y^{4} - 1000 x y^{6} + 400 y^{4} x^{2} + 1000 y^{6} x + 25 \cdot 25 y^{8}$

Multiply $25$ by $25$ .

$256 x^{4} + 640 x^{3} y^{2} + 400 x^{2} y^{4} - 640 x^{3} y^{2} - 1600 x^{2} y^{4} - 1000 x y^{6} + 400 y^{4} x^{2} + 1000 y^{6} x + 625 y^{8}$

Simplify by adding terms.

Combine the opposite terms in $256 x^{4} + 640 x^{3} y^{2} + 400 x^{2} y^{4} - 640 x^{3} y^{2} - 1600 x^{2} y^{4} - 1000 x y^{6} + 400 y^{4} x^{2} + 1000 y^{6} x + 625 y^{8}$ .

Subtract $640 x^{3} y^{2}$ from $640 x^{3} y^{2}$ .

$256 x^{4} + 400 x^{2} y^{4} + 0 - 1600 x^{2} y^{4} - 1000 x y^{6} + 400 y^{4} x^{2} + 1000 y^{6} x + 625 y^{8}$

Add $256 x^{4} + 400 x^{2} y^{4}$ and $0$ .

$256 x^{4} + 400 x^{2} y^{4} - 1600 x^{2} y^{4} - 1000 x y^{6} + 400 y^{4} x^{2} + 1000 y^{6} x + 625 y^{8}$

Reorder the factors in the terms $- 1000 x y^{6}$ and $1000 y^{6} x$ .

$256 x^{4} + 400 x^{2} y^{4} - 1600 x^{2} y^{4} - 1000 y^{6} x + 400 y^{4} x^{2} + 1000 y^{6} x + 625 y^{8}$

Add $- 1000 y^{6} x$ and $1000 y^{6} x$ .

$256 x^{4} + 400 x^{2} y^{4} - 1600 x^{2} y^{4} + 400 y^{4} x^{2} + 0 + 625 y^{8}$

Add $256 x^{4} + 400 x^{2} y^{4} - 1600 x^{2} y^{4} + 400 y^{4} x^{2}$ and $0$ .

$256 x^{4} + 400 x^{2} y^{4} - 1600 x^{2} y^{4} + 400 y^{4} x^{2} + 625 y^{8}$

Subtract $1600 x^{2} y^{4}$ from $400 x^{2} y^{4}$ .

$256 x^{4} - 1200 x^{2} y^{4} + 400 y^{4} x^{2} + 625 y^{8}$

Step 9

Add $- 1200 x^{2} y^{4}$ and $400 y^{4} x^{2}$ .

Move $y^{4}$ .

$256 x^{4} - 1200 x^{2} y^{4} + 400 x^{2} y^{4} + 625 y^{8}$

Add $- 1200 x^{2} y^{4}$ and $400 x^{2} y^{4}$ .

$256 x^{4} - 800 x^{2} y^{4} + 625 y^{8}$

Expand the Trigonometric Expression (4x-5y^2)^2(4x+5y^2)^2

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