Simplify square root of 3( square root of 3-1)^2

$\sqrt{3 {(\sqrt{3} - 1)}^{2}}$

Step 1

Rewrite $3 {(\sqrt{3} - 1)}^{2}$ as ${(\sqrt{3} + i^{2})}^{2} \cdot 3$ .

Reorder $3$ and ${(\sqrt{3} - 1)}^{2}$ .

$\sqrt{{(\sqrt{3} - 1)}^{2} \cdot 3}$

Rewrite $- 1$ as $i^{2}$ .

$\sqrt{{(\sqrt{3} + i^{2})}^{2} \cdot 3}$

Step 2

Pull terms out from under the radical.

$(\sqrt{3} + i^{2}) \sqrt{3}$

Step 3

Rewrite $i^{2}$ as $- 1$ .

$(\sqrt{3} - 1) \sqrt{3}$

Step 4

Simplify terms.

Apply the distributive property.

$\sqrt{3} \sqrt{3} - 1 \sqrt{3}$

Combine using the product rule for radicals.

$\sqrt{3 \cdot 3} - 1 \sqrt{3}$

Rewrite $- 1 \sqrt{3}$ as $- \sqrt{3}$ .

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

Step 5

Simplify each term.

Multiply $3$ by $3$ .

$\sqrt{9} - \sqrt{3}$

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

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

Pull terms out from under the radical, assuming positive real numbers.

$3 - \sqrt{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$3 - \sqrt{3}$

Decimal Form:

$1.26794919 \dots$