Evaluate square root of 12*(-1+ square root of 5)

$\sqrt{12} \cdot (- 1 + \sqrt{5})$

Step 1

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$\sqrt{4 (3)} \cdot (- 1 + \sqrt{5})$

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

$\sqrt{2^{2} \cdot 3} \cdot (- 1 + \sqrt{5})$

$\sqrt{2^{2} \cdot 3} \cdot (- 1 + \sqrt{5})$

Step 2

Pull terms out from under the radical.

$2 \sqrt{3} \cdot (- 1 + \sqrt{5})$

Step 3

Apply the distributive property.

$2 \sqrt{3} \cdot - 1 + 2 \sqrt{3} \sqrt{5}$

Step 4

Multiply $- 1$ by $2$ .

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

Step 5

Multiply $2 \sqrt{3} \sqrt{5}$ .

Combine using the product rule for radicals.

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

Multiply $5$ by $3$ .

$- 2 \sqrt{3} + 2 \sqrt{15}$

$- 2 \sqrt{3} + 2 \sqrt{15}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- 2 \sqrt{3} + 2 \sqrt{15}$

Decimal Form:

$4.28186507 \dots$

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