Algebra Examples

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

Step 1

Expand $(\sqrt{3} + \sqrt{6}) (\sqrt{2} - 2)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine using the product rule for radicals.

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

Step 2.1.2

Multiply $3$ by $2$ .

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

Step 2.1.3

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

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

Step 2.1.4

Combine using the product rule for radicals.

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

Step 2.1.5

Multiply $6$ by $2$ .

$\sqrt{6} - 2 \sqrt{3} + \sqrt{12} + \sqrt{6} \cdot - 2$

Step 2.1.6

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

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Step 2.1.6.1

Factor $4$ out of $12$ .

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

Step 2.1.6.2

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

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

Step 2.1.7

Pull terms out from under the radical.

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

Step 2.1.8

Move $- 2$ to the left of $\sqrt{6}$ .

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

Step 2.2

Subtract $2 \sqrt{6}$ from $\sqrt{6}$ .

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

Step 2.3

Add $- 2 \sqrt{3}$ and $2 \sqrt{3}$ .

$- \sqrt{6} + 0$

Step 2.4

Add $- \sqrt{6}$ and $0$ .

$- \sqrt{6}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- \sqrt{6}$

Decimal Form:

$- 2.44948974 \dots$