Root-ception

Algebra Level 2

Find the answer of $\sqrt { 5+2\left( \sqrt { 9+6\sqrt { 2 } } -\sqrt { 3 } \right) } -\sqrt { 2 }$

Note : ${ \left( a+b \right) }^{ 2 }={ a }^{ 2 }+2ab+{ b }^{ 2 }$

\sqrt { 6 }

-\sqrt { 3 }

-\sqrt { 6 }

\sqrt { 3 }

1 solution

Chan Lye Lee
Nov 1, 2015

Note that $9+6\sqrt{2} = 3\left(3+2\sqrt{2}\right)=3\left(\sqrt{2} +1\right)^2$ . Hence, $\sqrt{9+6\sqrt{2}}=\sqrt{3} \left(\sqrt{2} +1\right)$ and so $\sqrt{9+6\sqrt{2}} -\sqrt{3}=\sqrt{3} \sqrt{2}$ .

Next, $5+2\sqrt{3} \sqrt{2} = \left(\sqrt{2} +\sqrt{3}\right)^2$ . Hence, $\sqrt{5+2\sqrt{3} \sqrt{2}} = \sqrt{2} +\sqrt{3}$ and so $\sqrt { 5+2\left( \sqrt { 9+6\sqrt { 2 } } -\sqrt { 3 } \right) } -\sqrt { 2 } = \sqrt{3}$ .

Root-ception

1 solution

0 pending reports