Precalculus Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify each term.

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Step 2.1

Multiply $3$ by $- 3$ .

$3 \sqrt{6} - 9 - 2 \sqrt{2} \sqrt{6} - 2 \sqrt{2} \cdot - 3$

Step 2.2

Multiply $- 2 \sqrt{2} \sqrt{6}$ .

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Step 2.2.1

Combine using the product rule for radicals.

$3 \sqrt{6} - 9 - 2 \sqrt{6 \cdot 2} - 2 \sqrt{2} \cdot - 3$

Step 2.2.2

Multiply $6$ by $2$ .

$3 \sqrt{6} - 9 - 2 \sqrt{12} - 2 \sqrt{2} \cdot - 3$

Step 2.3

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

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Step 2.3.1

Factor $4$ out of $12$ .

$3 \sqrt{6} - 9 - 2 \sqrt{4 (3)} - 2 \sqrt{2} \cdot - 3$

Step 2.3.2

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

$3 \sqrt{6} - 9 - 2 \sqrt{2^{2} \cdot 3} - 2 \sqrt{2} \cdot - 3$

Step 2.4

Pull terms out from under the radical.

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

Step 2.5

Multiply $2$ by $- 2$ .

$3 \sqrt{6} - 9 - 4 \sqrt{3} - 2 \sqrt{2} \cdot - 3$

Step 2.6

Multiply $- 3$ by $- 2$ .

$3 \sqrt{6} - 9 - 4 \sqrt{3} + 6 \sqrt{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$3 \sqrt{6} - 9 - 4 \sqrt{3} + 6 \sqrt{2}$

Decimal Form:

$- 0.09445262 \dots$