Simplify square root of 18(2+ square root of 6)-2 square root of 3

$\sqrt{18} (2 + \sqrt{6}) - 2 \sqrt{3}$

Simplify each term.

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

Factor $9$ out of $18$ .

$\sqrt{9 (2)} (2 + \sqrt{6}) - 2 \sqrt{3}$

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

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

Pull terms out from under the radical.

$3 \sqrt{2} (2 + \sqrt{6}) - 2 \sqrt{3}$

Apply the distributive property.

$3 \sqrt{2} \cdot 2 + 3 \sqrt{2} \sqrt{6} - 2 \sqrt{3}$

Multiply $2$ by $3$ .

$6 \sqrt{2} + 3 \sqrt{2} \sqrt{6} - 2 \sqrt{3}$

Multiply $3 \sqrt{2} \sqrt{6}$ .

Combine using the product rule for radicals.

$6 \sqrt{2} + 3 \sqrt{6 \cdot 2} - 2 \sqrt{3}$

Multiply $6$ by $2$ .

$6 \sqrt{2} + 3 \sqrt{12} - 2 \sqrt{3}$

Simplify each term.

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

Factor $4$ out of $12$ .

$6 \sqrt{2} + 3 \sqrt{4 (3)} - 2 \sqrt{3}$

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

$6 \sqrt{2} + 3 \sqrt{2^{2} \cdot 3} - 2 \sqrt{3}$

Pull terms out from under the radical.

$6 \sqrt{2} + 3 (2 \sqrt{3}) - 2 \sqrt{3}$

Multiply $2$ by $3$ .

$6 \sqrt{2} + 6 \sqrt{3} - 2 \sqrt{3}$

Subtract $2 \sqrt{3}$ from $6 \sqrt{3}$ .

$6 \sqrt{2} + 4 \sqrt{3}$

The result can be shown in multiple forms.

Exact Form:

$6 \sqrt{2} + 4 \sqrt{3}$

Decimal Form:

$15.41348460 \dots$