Simplify 4x^2 square root of 18x^3y^7

$4 x^{2} \sqrt{18 x^{3} y^{7}}$

Step 1

Rewrite $18 x^{3} y^{7}$ as ${(3 x y^{3})}^{2} \cdot (2 x y)$ .

Factor $9$ out of $18$ .

$4 x^{2} \sqrt{9 (2) x^{3} y^{7}}$

Rewrite $9$ as $3^{2}$ .

$4 x^{2} \sqrt{3^{2} \cdot 2 x^{3} y^{7}}$

Factor out $x^{2}$ .

$4 x^{2} \sqrt{3^{2} \cdot 2 (x^{2} x) y^{7}}$

Factor out $y^{6}$ .

$4 x^{2} \sqrt{3^{2} \cdot 2 (x^{2} x) (y^{6} y)}$

Rewrite $y^{6}$ as ${(y^{3})}^{2}$ .

$4 x^{2} \sqrt{3^{2} \cdot 2 (x^{2} x) ({(y^{3})}^{2} y)}$

Move $x$ .

$4 x^{2} \sqrt{3^{2} \cdot 2 (x^{2}) {(y^{3})}^{2} x y}$

Move $2$ .

$4 x^{2} \sqrt{3^{2} (x^{2}) {(y^{3})}^{2} \cdot 2 x y}$

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

$4 x^{2} \sqrt{{(3 x y^{3})}^{2} \cdot 2 x y}$

Add parentheses.

$4 x^{2} \sqrt{{(3 x y^{3})}^{2} \cdot 2 (x y)}$

Add parentheses.

$4 x^{2} \sqrt{{(3 x y^{3})}^{2} \cdot (2 x y)}$

Step 2

Pull terms out from under the radical.

$4 x^{2} (3 x y^{3} \sqrt{2 x y})$

Step 3

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

Move $x$ .

$4 (x \cdot x^{2}) (3 y^{3} \sqrt{2 x y})$

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

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

$4 (x^{1} x^{2}) (3 y^{3} \sqrt{2 x y})$

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

$4 x^{1 + 2} (3 y^{3} \sqrt{2 x y})$

Add $1$ and $2$ .

$4 x^{3} (3 y^{3} \sqrt{2 x y})$

Step 4

Multiply $3$ by $4$ .

$12 x^{3} (y^{3} \sqrt{2 x y})$

Step 5

Reorder factors in $12 x^{3} (y^{3} \sqrt{2 x y})$ .

$12 x^{3} y^{3} \sqrt{2 x y}$