Simplify the Radical Expression ( square root of 12+6)(- square root of 8- square root of 2)

$(\sqrt{12} + 6) (- \sqrt{8} - \sqrt{2})$

Simplify each term.

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

Factor $4$ out of $12$ .

$(\sqrt{4 (3)} + 6) (- \sqrt{8} - \sqrt{2})$

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

$(\sqrt{2^{2} \cdot 3} + 6) (- \sqrt{8} - \sqrt{2})$

Pull terms out from under the radical.

$(| 2 | \sqrt{3} + 6) (- \sqrt{8} - \sqrt{2})$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$(2 \sqrt{3} + 6) (- \sqrt{8} - \sqrt{2})$

Simplify each term.

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

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

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

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

Pull terms out from under the radical.

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

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

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

Multiply $2$ by $- 1$ .

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

Subtract $\sqrt{2}$ from $- 2 \sqrt{2}$ .

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

Apply the distributive property.

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

Multiply $2 \sqrt{3} (- 3 \sqrt{2})$ .

Multiply $- 3$ by $2$ .

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

Combine using the product rule for radicals.

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

Multiply $2$ by $3$ .

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

Multiply $- 3$ by $6$ .

$- 6 \sqrt{6} - 18 \sqrt{2}$

The result can be shown in multiple forms.

Exact Form:

$- 6 \sqrt{6} - 18 \sqrt{2}$

Decimal Form:

$- 40.15278257 \dots$