Simplify ( square root of 2+2 square root of 8)(3 square root of 6- square root of 5)

$(\sqrt{2} + 2 \sqrt{8}) (3 \sqrt{6} - \sqrt{5})$

Simplify each term.

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

Factor $4$ out of $8$ .

$(\sqrt{2} + 2 \sqrt{4 (2)}) (3 \sqrt{6} - \sqrt{5})$

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

$(\sqrt{2} + 2 \sqrt{2^{2} \cdot 2}) (3 \sqrt{6} - \sqrt{5})$

Pull terms out from under the radical.

$(\sqrt{2} + 2 (2 \sqrt{2})) (3 \sqrt{6} - \sqrt{5})$

Multiply $2$ by $2$ .

$(\sqrt{2} + 4 \sqrt{2}) (3 \sqrt{6} - \sqrt{5})$

Simplify terms.

Add $\sqrt{2}$ and $4 \sqrt{2}$ .

$5 \sqrt{2} (3 \sqrt{6} - \sqrt{5})$

Apply the distributive property.

$5 \sqrt{2} (3 \sqrt{6}) + 5 \sqrt{2} (- \sqrt{5})$

Multiply $5 \sqrt{2} (3 \sqrt{6})$ .

Multiply $3$ by $5$ .

$15 \sqrt{2} \sqrt{6} + 5 \sqrt{2} (- \sqrt{5})$

Combine using the product rule for radicals.

$15 \sqrt{6 \cdot 2} + 5 \sqrt{2} (- \sqrt{5})$

Multiply $6$ by $2$ .

$15 \sqrt{12} + 5 \sqrt{2} (- \sqrt{5})$

Multiply $5 \sqrt{2} (- \sqrt{5})$ .

Multiply $- 1$ by $5$ .

$15 \sqrt{12} - 5 \sqrt{2} \sqrt{5}$

Combine using the product rule for radicals.

$15 \sqrt{12} - 5 \sqrt{5 \cdot 2}$

Multiply $5$ by $2$ .

$15 \sqrt{12} - 5 \sqrt{10}$

Simplify each term.

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

Factor $4$ out of $12$ .

$15 \sqrt{4 (3)} - 5 \sqrt{10}$

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

$15 \sqrt{2^{2} \cdot 3} - 5 \sqrt{10}$

Pull terms out from under the radical.

$15 (2 \sqrt{3}) - 5 \sqrt{10}$

Multiply $2$ by $15$ .

$30 \sqrt{3} - 5 \sqrt{10}$

The result can be shown in multiple forms.

Exact Form:

$30 \sqrt{3} - 5 \sqrt{10}$

Decimal Form:

$36.15013592 \dots$