Basic Math Examples

$(2 \sqrt{3 - \sqrt{12}}) \cdot (\sqrt{5 - 7 \sqrt{12}})$

Step 1

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

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Step 1.1

Factor $4$ out of $12$ .

$2 \sqrt{3 - \sqrt{4 (3)}} \cdot \sqrt{5 - 7 \sqrt{12}}$

Step 1.2

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

$2 \sqrt{3 - \sqrt{2^{2} \cdot 3}} \cdot \sqrt{5 - 7 \sqrt{12}}$

Step 2

Pull terms out from under the radical.

$2 \sqrt{3 - (2 \sqrt{3})} \cdot \sqrt{5 - 7 \sqrt{12}}$

Step 3

Multiply $2$ by $- 1$ .

$2 \sqrt{3 - 2 \sqrt{3}} \cdot \sqrt{5 - 7 \sqrt{12}}$

Step 4

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

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Step 4.1

Factor $4$ out of $12$ .

$2 \sqrt{3 - 2 \sqrt{3}} \cdot \sqrt{5 - 7 \sqrt{4 (3)}}$

Step 4.2

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

$2 \sqrt{3 - 2 \sqrt{3}} \cdot \sqrt{5 - 7 \sqrt{2^{2} \cdot 3}}$

Step 5

Pull terms out from under the radical.

$2 \sqrt{3 - 2 \sqrt{3}} \cdot \sqrt{5 - 7 (2 \sqrt{3})}$

Step 6

Multiply $2$ by $- 7$ .

$2 \sqrt{3 - 2 \sqrt{3}} \cdot \sqrt{5 - 14 \sqrt{3}}$

Step 7

Combine using the product rule for radicals.

$2 \sqrt{(5 - 14 \sqrt{3}) (3 - 2 \sqrt{3})}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{(5 - 14 \sqrt{3}) (3 - 2 \sqrt{3})}$

Decimal Form:

$5.97774472 \dots$