Simplify 1/3 square root of 18n^2* square root of 2n

$\frac{1}{3} \sqrt{18 n^{2}} \cdot \sqrt{2 n}$

Step 1

Rewrite $18 n^{2}$ as ${(3 n)}^{2} \cdot 2$ .

Factor $9$ out of $18$ .

$\frac{1}{3} \cdot \sqrt{9 (2) n^{2}} \cdot \sqrt{2 n}$

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

$\frac{1}{3} \cdot \sqrt{3^{2} \cdot 2 n^{2}} \cdot \sqrt{2 n}$

Move $2$ .

$\frac{1}{3} \cdot \sqrt{3^{2} n^{2} \cdot 2} \cdot \sqrt{2 n}$

Rewrite $3^{2} n^{2}$ as ${(3 n)}^{2}$ .

$\frac{1}{3} \cdot \sqrt{{(3 n)}^{2} \cdot 2} \cdot \sqrt{2 n}$

Step 2

Pull terms out from under the radical.

$\frac{1}{3} \cdot (3 n \sqrt{2}) \cdot \sqrt{2 n}$

Step 3

Cancel the common factor of $3$ .

Factor $3$ out of $3 n \sqrt{2}$ .

$\frac{1}{3} \cdot (3 (n \sqrt{2})) \cdot \sqrt{2 n}$

Cancel the common factor.

$\frac{1}{3} \cdot (3 (n \sqrt{2})) \cdot \sqrt{2 n}$

Rewrite the expression.

$n \sqrt{2} \cdot \sqrt{2 n}$

Step 4

Multiply $n \sqrt{2} \sqrt{2 n}$ .

Combine using the product rule for radicals.

$n \sqrt{2 n \cdot 2}$

Multiply $2$ by $2$ .

$n \sqrt{4 n}$

Step 5

Rewrite $4$ as $2^{2}$ .

$n \sqrt{2^{2} n}$

Step 6

Pull terms out from under the radical.

$n (2 \sqrt{n})$

Step 7

Rewrite using the commutative property of multiplication.

$2 n \sqrt{n}$