Find the Product xy^2(x^2+3xy+9)

$x y^{2} (x^{2} + 3 x y + 9)$

Apply the distributive property.

$x y^{2} x^{2} + x y^{2} (3 x y) + x y^{2} \cdot 9$

Simplify.

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

Move $x^{2}$ .

$x^{2} x y^{2} + x y^{2} (3 x y) + x y^{2} \cdot 9$

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

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

$x^{2} x^{1} y^{2} + x y^{2} (3 x y) + x y^{2} \cdot 9$

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

$x^{2 + 1} y^{2} + x y^{2} (3 x y) + x y^{2} \cdot 9$

Add $2$ and $1$ .

$x^{3} y^{2} + x y^{2} (3 x y) + x y^{2} \cdot 9$

Rewrite using the commutative property of multiplication.

$x^{3} y^{2} + 3 x y^{2} (x y) + x y^{2} \cdot 9$

Move $9$ to the left of $x y^{2}$ .

$x^{3} y^{2} + 3 x y^{2} (x y) + 9 \cdot (x y^{2})$

Simplify each term.

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

Move $x$ .

$x^{3} y^{2} + 3 (x \cdot x) y^{2} y + 9 \cdot (x y^{2})$

Multiply $x$ by $x$ .

$x^{3} y^{2} + 3 x^{2} y^{2} y + 9 \cdot (x y^{2})$

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

Move $y$ .

$x^{3} y^{2} + 3 x^{2} (y \cdot y^{2}) + 9 \cdot (x y^{2})$

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

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

$x^{3} y^{2} + 3 x^{2} (y^{1} y^{2}) + 9 \cdot (x y^{2})$

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

$x^{3} y^{2} + 3 x^{2} y^{1 + 2} + 9 \cdot (x y^{2})$

Add $1$ and $2$ .

$x^{3} y^{2} + 3 x^{2} y^{3} + 9 \cdot (x y^{2})$

$x^{3} y^{2} + 3 x^{2} y^{3} + 9 x y^{2}$