Simplify square root of 3(2- square root of 24)

$\sqrt{3} (2 - \sqrt{24})$

Step 1

Simplify each term.

Rewrite $24$ as $2^{2} \cdot 6$ .

Factor $4$ out of $24$ .

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

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 1$ .

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

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

Step 2

Simplify by multiplying through.

Apply the distributive property.

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

Move $2$ to the left of $\sqrt{3}$ .

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

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

Step 3

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

Combine using the product rule for radicals.

$2 \cdot \sqrt{3} - 2 \sqrt{3 \cdot 6}$

Multiply $3$ by $6$ .

$2 \cdot \sqrt{3} - 2 \sqrt{18}$

$2 \cdot \sqrt{3} - 2 \sqrt{18}$

Step 4

Simplify each term.

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

Factor $9$ out of $18$ .

$2 \sqrt{3} - 2 \sqrt{9 (2)}$

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

$2 \sqrt{3} - 2 \sqrt{3^{2} \cdot 2}$

$2 \sqrt{3} - 2 \sqrt{3^{2} \cdot 2}$

Pull terms out from under the radical.

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

Multiply $3$ by $- 2$ .

$2 \sqrt{3} - 6 \sqrt{2}$

$2 \sqrt{3} - 6 \sqrt{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{3} - 6 \sqrt{2}$

Decimal Form:

$- 5.02117975 \dots$

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