Simplify (3-4 square root of 10)( square root of 5+2 square root of 3)

$(3 - 4 \sqrt{10}) (\sqrt{5} + 2 \sqrt{3})$

Step 1

Expand $(3 - 4 \sqrt{10}) (\sqrt{5} + 2 \sqrt{3})$ using the FOIL Method.

Apply the distributive property.

$3 (\sqrt{5} + 2 \sqrt{3}) - 4 \sqrt{10} (\sqrt{5} + 2 \sqrt{3})$

Apply the distributive property.

$3 \sqrt{5} + 3 (2 \sqrt{3}) - 4 \sqrt{10} (\sqrt{5} + 2 \sqrt{3})$

Apply the distributive property.

$3 \sqrt{5} + 3 (2 \sqrt{3}) - 4 \sqrt{10} \sqrt{5} - 4 \sqrt{10} (2 \sqrt{3})$

Step 2

Simplify each term.

Multiply $2$ by $3$ .

$3 \sqrt{5} + 6 \sqrt{3} - 4 \sqrt{10} \sqrt{5} - 4 \sqrt{10} (2 \sqrt{3})$

Multiply $- 4 \sqrt{10} \sqrt{5}$ .

Combine using the product rule for radicals.

$3 \sqrt{5} + 6 \sqrt{3} - 4 \sqrt{5 \cdot 10} - 4 \sqrt{10} (2 \sqrt{3})$

Multiply $5$ by $10$ .

$3 \sqrt{5} + 6 \sqrt{3} - 4 \sqrt{50} - 4 \sqrt{10} (2 \sqrt{3})$

Rewrite $50$ as $5^{2} \cdot 2$ .

Factor $25$ out of $50$ .

$3 \sqrt{5} + 6 \sqrt{3} - 4 \sqrt{25 (2)} - 4 \sqrt{10} (2 \sqrt{3})$

Rewrite $25$ as $5^{2}$ .

$3 \sqrt{5} + 6 \sqrt{3} - 4 \sqrt{5^{2} \cdot 2} - 4 \sqrt{10} (2 \sqrt{3})$

Pull terms out from under the radical.

$3 \sqrt{5} + 6 \sqrt{3} - 4 (5 \sqrt{2}) - 4 \sqrt{10} (2 \sqrt{3})$

Multiply $5$ by $- 4$ .

$3 \sqrt{5} + 6 \sqrt{3} - 20 \sqrt{2} - 4 \sqrt{10} (2 \sqrt{3})$

Multiply $- 4 \sqrt{10} (2 \sqrt{3})$ .

Multiply $2$ by $- 4$ .

$3 \sqrt{5} + 6 \sqrt{3} - 20 \sqrt{2} - 8 \sqrt{10} \sqrt{3}$

Combine using the product rule for radicals.

$3 \sqrt{5} + 6 \sqrt{3} - 20 \sqrt{2} - 8 \sqrt{3 \cdot 10}$

Multiply $3$ by $10$ .

$3 \sqrt{5} + 6 \sqrt{3} - 20 \sqrt{2} - 8 \sqrt{30}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$3 \sqrt{5} + 6 \sqrt{3} - 20 \sqrt{2} - 8 \sqrt{30}$

Decimal Form:

$- 55.00156706 \dots$