Simplify square root of 15n^2* square root of 10n^3

$\sqrt{15 n^{2}} \cdot \sqrt{10 n^{3}}$

Step 1

Reorder $15$ and $n^{2}$ .

$\sqrt{n^{2} \cdot 15} \cdot \sqrt{10 n^{3}}$

Step 2

Pull terms out from under the radical.

$n \sqrt{15} \cdot \sqrt{10 n^{3}}$

Step 3

Rewrite $10 n^{3}$ as $n^{2} \cdot (10 n)$ .

Factor out $n^{2}$ .

$n \sqrt{15} \cdot \sqrt{10 (n^{2} n)}$

Reorder $10$ and $n^{2}$ .

$n \sqrt{15} \cdot \sqrt{n^{2} \cdot 10 n}$

Add parentheses.

$n \sqrt{15} \cdot \sqrt{n^{2} \cdot (10 n)}$

$n \sqrt{15} \cdot \sqrt{n^{2} \cdot (10 n)}$

Step 4

Pull terms out from under the radical.

$n \sqrt{15} \cdot (n \sqrt{10 n})$

Step 5

Multiply $n$ by $n$ by adding the exponents.

Move $n$ .

$n \cdot n \sqrt{15} \cdot \sqrt{10 n}$

Multiply $n$ by $n$ .

$n^{2} \sqrt{15} \cdot \sqrt{10 n}$

$n^{2} \sqrt{15} \cdot \sqrt{10 n}$

Step 6

Multiply $n^{2} \sqrt{15} \sqrt{10 n}$ .

Combine using the product rule for radicals.

$n^{2} \sqrt{10 n \cdot 15}$

Multiply $15$ by $10$ .

$n^{2} \sqrt{150 n}$

$n^{2} \sqrt{150 n}$

Step 7

Rewrite $150 n$ as $5^{2} \cdot (6 n)$ .

Factor $25$ out of $150$ .

$n^{2} \sqrt{25 (6) n}$

Rewrite $25$ as $5^{2}$ .

$n^{2} \sqrt{5^{2} \cdot 6 n}$

Add parentheses.

$n^{2} \sqrt{5^{2} \cdot (6 n)}$

$n^{2} \sqrt{5^{2} \cdot (6 n)}$

Step 8

Pull terms out from under the radical.

$n^{2} (5 \sqrt{6 n})$

Step 9

Rewrite using the commutative property of multiplication.

$5 n^{2} \sqrt{6 n}$

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