Simplify square root of 3( square root of 6-2 square root of 54)

$\sqrt{3} (\sqrt{6} - 2 \sqrt{54})$

Step 1

Simplify each term.

Rewrite $54$ as $3^{2} \cdot 6$ .

Factor $9$ out of $54$ .

$\sqrt{3} (\sqrt{6} - 2 \sqrt{9 (6)})$

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

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

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

Pull terms out from under the radical.

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

Multiply $3$ by $- 2$ .

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

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

Step 2

Subtract $6 \sqrt{6}$ from $\sqrt{6}$ .

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

Step 3

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

Combine using the product rule for radicals.

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

Multiply $3$ by $6$ .

$- 5 \sqrt{18}$

$- 5 \sqrt{18}$

Step 4

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

Factor $9$ out of $18$ .

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

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

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

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

Step 5

Pull terms out from under the radical.

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

Step 6

Multiply $3$ by $- 5$ .

$- 15 \sqrt{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- 15 \sqrt{2}$

Decimal Form:

$- 21.21320343 \dots$

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