Simplify the Radical Expression square root of 3(-5 square root of 10+ square root of 6)

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

Apply the distributive property.

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

Multiply $\sqrt{3} (- 5 \sqrt{10})$ .

Combine using the product rule for radicals.

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

Multiply $3$ by $10$ .

$- 5 \sqrt{30} + \sqrt{3} \sqrt{6}$

Combine using the product rule for radicals.

$- 5 \sqrt{30} + \sqrt{3 \cdot 6}$

Simplify each term.

Multiply $3$ by $6$ .

$- 5 \sqrt{30} + \sqrt{18}$

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

Factor $9$ out of $18$ .

$- 5 \sqrt{30} + \sqrt{9 (2)}$

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

$- 5 \sqrt{30} + \sqrt{3^{2} \cdot 2}$

Pull terms out from under the radical.

$- 5 \sqrt{30} + | 3 | \sqrt{2}$

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$- 5 \sqrt{30} + 3 \sqrt{2}$

The result can be shown in multiple forms.

Exact Form:

$- 5 \sqrt{30} + 3 \sqrt{2}$

Decimal Form:

$- 23.14348718 \dots$