Simplify (3 square root of 3-2 square root of 2)* square root of 6

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

Step 1

Apply the distributive property.

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

Step 2

Multiply $3 \sqrt{3} \sqrt{6}$ .

Combine using the product rule for radicals.

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

Multiply $6$ by $3$ .

$3 \sqrt{18} - 2 \sqrt{2} \sqrt{6}$

$3 \sqrt{18} - 2 \sqrt{2} \sqrt{6}$

Step 3

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

Combine using the product rule for radicals.

$3 \sqrt{18} - 2 \sqrt{6 \cdot 2}$

Multiply $6$ by $2$ .

$3 \sqrt{18} - 2 \sqrt{12}$

$3 \sqrt{18} - 2 \sqrt{12}$

Step 4

Simplify each term.

Rewrite $18$ as $3^{2} \cdot 2$ .

Factor $9$ out of $18$ .

$3 \sqrt{9 (2)} - 2 \sqrt{12}$

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

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

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

Pull terms out from under the radical.

$3 (3 \sqrt{2}) - 2 \sqrt{12}$

Multiply $3$ by $3$ .

$9 \sqrt{2} - 2 \sqrt{12}$

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

Factor $4$ out of $12$ .

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

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 2$ .

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

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$5.79971883 \dots$

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