Evaluate 3 square root of 18+3 square root of 12+2 square root of 27

$3 \sqrt{18} + 3 \sqrt{12} + 2 \sqrt{27}$

Step 1

Simplify each term.

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

Factor $9$ out of $18$ .

$3 \sqrt{9 (2)} + 3 \sqrt{12} + 2 \sqrt{27}$

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

$3 \sqrt{3^{2} \cdot 2} + 3 \sqrt{12} + 2 \sqrt{27}$

Pull terms out from under the radical.

$3 (3 \sqrt{2}) + 3 \sqrt{12} + 2 \sqrt{27}$

Multiply $3$ by $3$ .

$9 \sqrt{2} + 3 \sqrt{12} + 2 \sqrt{27}$

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

Factor $4$ out of $12$ .

$9 \sqrt{2} + 3 \sqrt{4 (3)} + 2 \sqrt{27}$

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

$9 \sqrt{2} + 3 \sqrt{2^{2} \cdot 3} + 2 \sqrt{27}$

Pull terms out from under the radical.

$9 \sqrt{2} + 3 (2 \sqrt{3}) + 2 \sqrt{27}$

Multiply $2$ by $3$ .

$9 \sqrt{2} + 6 \sqrt{3} + 2 \sqrt{27}$

Rewrite $27$ as $3^{2} \cdot 3$ .

Factor $9$ out of $27$ .

$9 \sqrt{2} + 6 \sqrt{3} + 2 \sqrt{9 (3)}$

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

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

Pull terms out from under the radical.

$9 \sqrt{2} + 6 \sqrt{3} + 2 (3 \sqrt{3})$

Multiply $3$ by $2$ .

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

Step 2

Add $6 \sqrt{3}$ and $6 \sqrt{3}$ .

$9 \sqrt{2} + 12 \sqrt{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$9 \sqrt{2} + 12 \sqrt{3}$

Decimal Form:

$33.51253175 \dots$