Simplify 3 square root of 12+2 square root of 108-3 square root of 48

$3 \sqrt{12 + 2 \sqrt{108 - 3 \sqrt{48}}}$

Rewrite $48$ as $4^{2} \cdot 3$ .

Factor $16$ out of $48$ .

$3 \sqrt{12 + 2 \sqrt{108 - 3 \sqrt{16 (3)}}}$

Rewrite $16$ as $4^{2}$ .

$3 \sqrt{12 + 2 \sqrt{108 - 3 \sqrt{4^{2} \cdot 3}}}$

Pull terms out from under the radical.

$3 \sqrt{12 + 2 \sqrt{108 - 3 (4 \sqrt{3})}}$

Multiply $4$ by $- 3$ .

$3 \sqrt{12 + 2 \sqrt{108 - 12 \sqrt{3}}}$

The result can be shown in multiple forms.

Exact Form:

$3 \sqrt{12 + 2 \sqrt{108 - 12 \sqrt{3}}}$

Decimal Form:

$16.61627289 \dots$