Simplify the Radical Expression -3 square root of 3(2+ square root of 6)

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

Apply the distributive property.

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

Multiply $2$ by $- 3$ .

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

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

Combine using the product rule for radicals.

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

Multiply $6$ by $3$ .

$- 6 \sqrt{3} - 3 \sqrt{18}$

Simplify each term.

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

Factor $9$ out of $18$ .

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

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

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

Pull terms out from under the radical.

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

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

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

Multiply $3$ by $- 3$ .

$- 6 \sqrt{3} - 9 \sqrt{2}$

The result can be shown in multiple forms.

Exact Form:

$- 6 \sqrt{3} - 9 \sqrt{2}$

Decimal Form:

$- 23.12022690 \dots$