میدانیم $\sqrt[n]{{a \times b}} = \sqrt[n]{a} \times \sqrt[n]{b}$
تک تک عبارات رادیکالی را باز میکنیم.
$\sqrt {12} = \sqrt {4 \times 3} = \sqrt 4 \times \sqrt 3 = 2 \times \sqrt 3 = 2\sqrt 3 $
$\sqrt {147} = \sqrt {49 \times 3} = \sqrt {49} \times \sqrt 3 = 7 \times \sqrt 3 = 7\sqrt 3 $
$\sqrt {48} = \sqrt {16 \times 3} = \sqrt {16} \times \sqrt 3 = 4 \times \sqrt 3 = 4\sqrt 3 $
$\frac{{3\sqrt {12} - \sqrt {147} + \sqrt {48} }}{{3\sqrt 6 }} = \frac{{6\sqrt 3 - 7\sqrt 3 + 4\sqrt 3 }}{{3\sqrt 6 }} = \frac{{\mathop {\cancel{3}}\limits^1 \sqrt 3 }}{{\cancel{3}\sqrt 6 }} = \frac{{ - \mathop {\cancel{{\sqrt 3 }}}\limits^1 }}{{\sqrt 2 \times \cancel{{\sqrt 3 }}}} = \frac{{ - 1}}{{\sqrt 2 }} \times \frac{{\sqrt 2 }}{{\sqrt 2 }} = - \frac{{\sqrt 2 }}{2}$