Find the Product (2 square root of 7+3 square root of 6)(5 square root of 2+4 square root of 3)

$(2 \sqrt{7} + 3 \sqrt{6}) (5 \sqrt{2} + 4 \sqrt{3})$

Step 1

Expand $(2 \sqrt{7} + 3 \sqrt{6}) (5 \sqrt{2} + 4 \sqrt{3})$ using the FOIL Method.

Apply the distributive property.

$2 \sqrt{7} (5 \sqrt{2} + 4 \sqrt{3}) + 3 \sqrt{6} (5 \sqrt{2} + 4 \sqrt{3})$

Apply the distributive property.

$2 \sqrt{7} (5 \sqrt{2}) + 2 \sqrt{7} (4 \sqrt{3}) + 3 \sqrt{6} (5 \sqrt{2} + 4 \sqrt{3})$

Apply the distributive property.

$2 \sqrt{7} (5 \sqrt{2}) + 2 \sqrt{7} (4 \sqrt{3}) + 3 \sqrt{6} (5 \sqrt{2}) + 3 \sqrt{6} (4 \sqrt{3})$

Step 2

Simplify each term.

Multiply $2 \sqrt{7} (5 \sqrt{2})$ .

Multiply $5$ by $2$ .

$10 \sqrt{7} \sqrt{2} + 2 \sqrt{7} (4 \sqrt{3}) + 3 \sqrt{6} (5 \sqrt{2}) + 3 \sqrt{6} (4 \sqrt{3})$

Combine using the product rule for radicals.

$10 \sqrt{2 \cdot 7} + 2 \sqrt{7} (4 \sqrt{3}) + 3 \sqrt{6} (5 \sqrt{2}) + 3 \sqrt{6} (4 \sqrt{3})$

Multiply $2$ by $7$ .

$10 \sqrt{14} + 2 \sqrt{7} (4 \sqrt{3}) + 3 \sqrt{6} (5 \sqrt{2}) + 3 \sqrt{6} (4 \sqrt{3})$

Multiply $2 \sqrt{7} (4 \sqrt{3})$ .

Multiply $4$ by $2$ .

$10 \sqrt{14} + 8 \sqrt{7} \sqrt{3} + 3 \sqrt{6} (5 \sqrt{2}) + 3 \sqrt{6} (4 \sqrt{3})$

Combine using the product rule for radicals.

$10 \sqrt{14} + 8 \sqrt{3 \cdot 7} + 3 \sqrt{6} (5 \sqrt{2}) + 3 \sqrt{6} (4 \sqrt{3})$

Multiply $3$ by $7$ .

$10 \sqrt{14} + 8 \sqrt{21} + 3 \sqrt{6} (5 \sqrt{2}) + 3 \sqrt{6} (4 \sqrt{3})$

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

Multiply $5$ by $3$ .

$10 \sqrt{14} + 8 \sqrt{21} + 15 \sqrt{6} \sqrt{2} + 3 \sqrt{6} (4 \sqrt{3})$

Combine using the product rule for radicals.

$10 \sqrt{14} + 8 \sqrt{21} + 15 \sqrt{2 \cdot 6} + 3 \sqrt{6} (4 \sqrt{3})$

Multiply $2$ by $6$ .

$10 \sqrt{14} + 8 \sqrt{21} + 15 \sqrt{12} + 3 \sqrt{6} (4 \sqrt{3})$

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

Factor $4$ out of $12$ .

$10 \sqrt{14} + 8 \sqrt{21} + 15 \sqrt{4 (3)} + 3 \sqrt{6} (4 \sqrt{3})$

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

$10 \sqrt{14} + 8 \sqrt{21} + 15 \sqrt{2^{2} \cdot 3} + 3 \sqrt{6} (4 \sqrt{3})$

Pull terms out from under the radical.

$10 \sqrt{14} + 8 \sqrt{21} + 15 (2 \sqrt{3}) + 3 \sqrt{6} (4 \sqrt{3})$

Multiply $2$ by $15$ .

$10 \sqrt{14} + 8 \sqrt{21} + 30 \sqrt{3} + 3 \sqrt{6} (4 \sqrt{3})$

Multiply $3 \sqrt{6} (4 \sqrt{3})$ .

Multiply $4$ by $3$ .

$10 \sqrt{14} + 8 \sqrt{21} + 30 \sqrt{3} + 12 \sqrt{6} \sqrt{3}$

Combine using the product rule for radicals.

$10 \sqrt{14} + 8 \sqrt{21} + 30 \sqrt{3} + 12 \sqrt{3 \cdot 6}$

Multiply $3$ by $6$ .

$10 \sqrt{14} + 8 \sqrt{21} + 30 \sqrt{3} + 12 \sqrt{18}$

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

Factor $9$ out of $18$ .

$10 \sqrt{14} + 8 \sqrt{21} + 30 \sqrt{3} + 12 \sqrt{9 (2)}$

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

$10 \sqrt{14} + 8 \sqrt{21} + 30 \sqrt{3} + 12 \sqrt{3^{2} \cdot 2}$

Pull terms out from under the radical.

$10 \sqrt{14} + 8 \sqrt{21} + 30 \sqrt{3} + 12 (3 \sqrt{2})$

Multiply $3$ by $12$ .

$10 \sqrt{14} + 8 \sqrt{21} + 30 \sqrt{3} + 36 \sqrt{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$10 \sqrt{14} + 8 \sqrt{21} + 30 \sqrt{3} + 36 \sqrt{2}$

Decimal Form:

$176.95039189 \dots$

Find the Product (2 square root of 7+3 square root of 6)(5 square root of 2+4 square root of 3)

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