Simplify square root of 6n(7n^3+ square root of 12)

$\sqrt{6 n} (7 n^{3} + \sqrt{12})$

Step 1

Simplify each term.

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$\sqrt{6 n} (7 n^{3} + \sqrt{4 (3)})$

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

$\sqrt{6 n} (7 n^{3} + \sqrt{2^{2} \cdot 3})$

$\sqrt{6 n} (7 n^{3} + \sqrt{2^{2} \cdot 3})$

Pull terms out from under the radical.

$\sqrt{6 n} (7 n^{3} + 2 \sqrt{3})$

$\sqrt{6 n} (7 n^{3} + 2 \sqrt{3})$

Step 2

Simplify by multiplying through.

Apply the distributive property.

$\sqrt{6 n} (7 n^{3}) + \sqrt{6 n} (2 \sqrt{3})$

Rewrite using the commutative property of multiplication.

$7 \sqrt{6 n} n^{3} + \sqrt{6 n} (2 \sqrt{3})$

$7 \sqrt{6 n} n^{3} + \sqrt{6 n} (2 \sqrt{3})$

Step 3

Multiply $\sqrt{6 n} (2 \sqrt{3})$ .

Combine using the product rule for radicals.

$7 \sqrt{6 n} n^{3} + 2 \sqrt{6 n \cdot 3}$

Multiply $3$ by $6$ .

$7 \sqrt{6 n} n^{3} + 2 \sqrt{18 n}$

$7 \sqrt{6 n} n^{3} + 2 \sqrt{18 n}$

Step 4

Simplify each term.

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

Factor $9$ out of $18$ .

$7 \sqrt{6 n} n^{3} + 2 \sqrt{9 (2) n}$

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

$7 \sqrt{6 n} n^{3} + 2 \sqrt{3^{2} \cdot 2 n}$

Add parentheses.

$7 \sqrt{6 n} n^{3} + 2 \sqrt{3^{2} \cdot (2 n)}$

$7 \sqrt{6 n} n^{3} + 2 \sqrt{3^{2} \cdot (2 n)}$

Pull terms out from under the radical.

$7 \sqrt{6 n} n^{3} + 2 (3 \sqrt{2 n})$

Multiply $3$ by $2$ .

$7 \sqrt{6 n} n^{3} + 6 \sqrt{2 n}$

$7 \sqrt{6 n} n^{3} + 6 \sqrt{2 n}$

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