Find the Product (x^5-3ay^2)^2

${(x^{5} - 3 a y^{2})}^{2}$

Step 1

Rewrite ${(x^{5} - 3 a y^{2})}^{2}$ as $(x^{5} - 3 a y^{2}) (x^{5} - 3 a y^{2})$ .

$(x^{5} - 3 a y^{2}) (x^{5} - 3 a y^{2})$

Step 2

Expand $(x^{5} - 3 a y^{2}) (x^{5} - 3 a y^{2})$ using the FOIL Method.

Apply the distributive property.

$x^{5} (x^{5} - 3 a y^{2}) - 3 a y^{2} (x^{5} - 3 a y^{2})$

Apply the distributive property.

$x^{5} x^{5} + x^{5} (- 3 a y^{2}) - 3 a y^{2} (x^{5} - 3 a y^{2})$

Apply the distributive property.

$x^{5} x^{5} + x^{5} (- 3 a y^{2}) - 3 a y^{2} x^{5} - 3 a y^{2} (- 3 a y^{2})$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $x^{5}$ by $x^{5}$ by adding the exponents.

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

$x^{5 + 5} + x^{5} (- 3 a y^{2}) - 3 a y^{2} x^{5} - 3 a y^{2} (- 3 a y^{2})$

Add $5$ and $5$ .

$x^{10} + x^{5} (- 3 a y^{2}) - 3 a y^{2} x^{5} - 3 a y^{2} (- 3 a y^{2})$

Rewrite using the commutative property of multiplication.

$x^{10} - 3 x^{5} a y^{2} - 3 a y^{2} x^{5} - 3 a y^{2} (- 3 a y^{2})$

Multiply $a$ by $a$ by adding the exponents.

Move $a$ .

$x^{10} - 3 x^{5} a y^{2} - 3 a y^{2} x^{5} - 3 (a \cdot a) y^{2} (- 3 y^{2})$

Multiply $a$ by $a$ .

$x^{10} - 3 x^{5} a y^{2} - 3 a y^{2} x^{5} - 3 a^{2} y^{2} (- 3 y^{2})$

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

Move $y^{2}$ .

$x^{10} - 3 x^{5} a y^{2} - 3 a y^{2} x^{5} - 3 a^{2} (y^{2} y^{2}) \cdot - 3$

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

$x^{10} - 3 x^{5} a y^{2} - 3 a y^{2} x^{5} - 3 a^{2} y^{2 + 2} \cdot - 3$

Add $2$ and $2$ .

$x^{10} - 3 x^{5} a y^{2} - 3 a y^{2} x^{5} - 3 a^{2} y^{4} \cdot - 3$

Multiply $- 3$ by $- 3$ .

$x^{10} - 3 x^{5} a y^{2} - 3 a y^{2} x^{5} + 9 a^{2} y^{4}$

Move $y^{2}$ .

$x^{10} - 3 a x^{5} y^{2} - 3 a x^{5} y^{2} + 9 a^{2} y^{4}$

Subtract $3 a x^{5} y^{2}$ from $- 3 a x^{5} y^{2}$ .

$x^{10} - 6 a x^{5} y^{2} + 9 a^{2} y^{4}$