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

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

Apply the distributive property.

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

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

Multiply $5$ by $3$ .

$15 \sqrt{2} \sqrt{6} + 3 \sqrt{2} (- 7 \sqrt{3})$

Combine using the product rule for radicals.

$15 \sqrt{6 \cdot 2} + 3 \sqrt{2} (- 7 \sqrt{3})$

Multiply $6$ by $2$ .

$15 \sqrt{12} + 3 \sqrt{2} (- 7 \sqrt{3})$

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

Multiply $- 7$ by $3$ .

$15 \sqrt{12} - 21 \sqrt{2} \sqrt{3}$

Combine using the product rule for radicals.

$15 \sqrt{12} - 21 \sqrt{3 \cdot 2}$

Multiply $3$ by $2$ .

$15 \sqrt{12} - 21 \sqrt{6}$

Simplify each term.

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

Factor $4$ out of $12$ .

$15 \sqrt{4 (3)} - 21 \sqrt{6}$

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

$15 \sqrt{2^{2} \cdot 3} - 21 \sqrt{6}$

Pull terms out from under the radical.

$15 (2 \sqrt{3}) - 21 \sqrt{6}$

Multiply $2$ by $15$ .

$30 \sqrt{3} - 21 \sqrt{6}$

The result can be shown in multiple forms.

Exact Form:

$30 \sqrt{3} - 21 \sqrt{6}$

Decimal Form:

$0.52223962 \dots$