Question

Fourth roots of $193-4\sqrt{2178}$ is

• A. $(7-\sqrt{2})$
• B. $(5-\sqrt{2})$
• C. $(3-\sqrt{2})$
• D. $(10-\sqrt{7})$

Answer 1

The correct option is C $(3-\sqrt{2})$

According to Question

$(193-4\sqrt{2178}{)}^{1/4}$

$=(193-4\sqrt{11\times 11\times 3\times 3\times 2}{)}^{1/4}$

$=(193-4\times 11\times 3\sqrt{2}{)}^{1/4}$

$=(121+72-132\sqrt{2}{)}^{1/4}$

$=({11}^2+(6\sqrt{2}{)}^2-2\times 11\times 6\sqrt{2}{)}^{1/4}$ $Using{a}^2+b^2-2ab=(a-b{)}^2$

$=(11-6\sqrt{2}{)}^{2\times 0.25}$

$=(9+2-6\sqrt{2}{)}^{0.5}$

$=(3^2+\sqrt{2}{}^2-2\times 3\times \sqrt{2}{)}^{0.5}$ $Using{a}^2+b^2-2ab=(a-b{)}^2$

$=(3-\sqrt{2}{)}^{0.5\times 2}$

$=3-\sqrt{2}$

$C$ is the right answer