Simplify 3 square root of 10(5 square root of 2-6 square root of 8)

$3 \sqrt{10} (5 \sqrt{2} - 6 \sqrt{8})$

Simplify each term.

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

Factor $4$ out of $8$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 6$ .

$3 \sqrt{10} (5 \sqrt{2} - 12 \sqrt{2})$

Subtract $12 \sqrt{2}$ from $5 \sqrt{2}$ .

$3 \sqrt{10} (- 7 \sqrt{2})$

Multiply $3 \sqrt{10} (- 7 \sqrt{2})$ .

Multiply $- 7$ by $3$ .

$- 21 \sqrt{10} \sqrt{2}$

Combine using the product rule for radicals.

$- 21 \sqrt{2 \cdot 10}$

Multiply $2$ by $10$ .

$- 21 \sqrt{20}$

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$- 21 \sqrt{4 (5)}$

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

$- 21 \sqrt{2^{2} \cdot 5}$

Pull terms out from under the radical.

$- 21 (2 \sqrt{5})$

Multiply $2$ by $- 21$ .

$- 42 \sqrt{5}$

The result can be shown in multiple forms.

Exact Form:

$- 42 \sqrt{5}$

Decimal Form:

$- 93.91485505 \dots$