Algebra Examples

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

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.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

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.

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite using the commutative property of multiplication.

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

Step 3.1.2

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

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Step 3.1.2.1

Move $x$ .

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

Step 3.1.2.2

Multiply $x$ by $x$ .

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

Step 3.1.3

Multiply $4$ by $4$ .

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

Step 3.1.4

Rewrite using the commutative property of multiplication.

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

Step 3.1.5

Multiply $4$ by $- 5$ .

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

Step 3.1.6

Rewrite using the commutative property of multiplication.

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

Step 3.1.7

Multiply $- 5$ by $4$ .

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

Step 3.1.8

Rewrite using the commutative property of multiplication.

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

Step 3.1.9

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

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Step 3.1.9.1

Move $y^{2}$ .

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

Step 3.1.9.2

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

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

Step 3.1.9.3

Add $2$ and $2$ .

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

Step 3.1.10

Multiply $- 5$ by $- 5$ .

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

Step 3.2

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

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Step 3.2.1

Move $y^{2}$ .

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

Step 3.2.2

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.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

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.

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Step 6.1

Simplify each term.

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Step 6.1.1

Rewrite using the commutative property of multiplication.

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

Step 6.1.2

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

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Step 6.1.2.1

Move $x$ .

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

Step 6.1.2.2

Multiply $x$ by $x$ .

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

Step 6.1.3

Multiply $4$ by $4$ .

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

Step 6.1.4

Rewrite using the commutative property of multiplication.

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

Step 6.1.5

Multiply $4$ by $5$ .

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

Step 6.1.6

Rewrite using the commutative property of multiplication.

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

Step 6.1.7

Multiply $5$ by $4$ .

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

Step 6.1.8

Rewrite using the commutative property of multiplication.

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

Step 6.1.9

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

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Step 6.1.9.1

Move $y^{2}$ .

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

Step 6.1.9.2

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

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

Step 6.1.9.3

Add $2$ and $2$ .

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

Step 6.1.10

Multiply $5$ by $5$ .

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

Step 6.2

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

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Step 6.2.1

Move $y^{2}$ .

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

Step 6.2.2

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.

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Step 8.1

Simplify each term.

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Step 8.1.1

Rewrite using the commutative property of multiplication.

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

Step 8.1.2

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

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Step 8.1.2.1

Move $x^{2}$ .

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

Step 8.1.2.2

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

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

Step 8.1.2.3

Add $2$ and $2$ .

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

Step 8.1.3

Multiply $16$ by $16$ .

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

Step 8.1.4

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

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Step 8.1.4.1

Move $x$ .

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

Step 8.1.4.2

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

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Step 8.1.4.2.1

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

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

Step 8.1.4.2.2

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

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

Step 8.1.4.3

Add $1$ and $2$ .

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

Step 8.1.5

Rewrite using the commutative property of multiplication.

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

Step 8.1.6

Multiply $16$ by $40$ .

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

Step 8.1.7

Rewrite using the commutative property of multiplication.

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

Step 8.1.8

Multiply $16$ by $25$ .

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

Step 8.1.9

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

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Step 8.1.9.1

Move $x^{2}$ .

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

Step 8.1.9.2

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

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Step 8.1.9.2.1

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

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

Step 8.1.9.2.2

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

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

Step 8.1.9.3

Add $2$ and $1$ .

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

Step 8.1.10

Multiply $16$ by $- 40$ .

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

Step 8.1.11

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

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Step 8.1.11.1

Move $x$ .

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

Step 8.1.11.2

Multiply $x$ by $x$ .

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

Step 8.1.12

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

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Step 8.1.12.1

Move $y^{2}$ .

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

Step 8.1.12.2

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

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

Step 8.1.12.3

Add $2$ and $2$ .

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

Step 8.1.13

Multiply $40$ by $- 40$ .

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

Step 8.1.14

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

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Step 8.1.14.1

Move $y^{4}$ .

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

Step 8.1.14.2

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

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

Step 8.1.14.3

Add $4$ and $2$ .

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

Step 8.1.15

Multiply $25$ by $- 40$ .

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

Step 8.1.16

Rewrite using the commutative property of multiplication.

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

Step 8.1.17

Multiply $25$ by $16$ .

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

Step 8.1.18

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

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Step 8.1.18.1

Move $y^{2}$ .

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

Step 8.1.18.2

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

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

Step 8.1.18.3

Add $2$ and $4$ .

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

Step 8.1.19

Rewrite using the commutative property of multiplication.

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

Step 8.1.20

Multiply $25$ by $40$ .

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

Step 8.1.21

Rewrite using the commutative property of multiplication.

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

Step 8.1.22

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

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Step 8.1.22.1

Move $y^{4}$ .

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

Step 8.1.22.2

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

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

Step 8.1.22.3

Add $4$ and $4$ .

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

Step 8.1.23

Multiply $25$ by $25$ .

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

Step 8.2

Simplify by adding terms.

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Step 8.2.1

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

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Step 8.2.1.1

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

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

Step 8.2.1.2

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

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

Step 8.2.1.3

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

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

Step 8.2.1.4

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

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

Step 8.2.1.5

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

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

Step 8.2.2

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 $400 y^{4} x^{2}$ .

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Step 9.1

Move $y^{4}$ .

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

Step 9.2

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

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