Evaluate: sqrt(6+2\sqrt{3)+2sqrt(2)+2sqrt(6)}-(1)/(sqrt(5-2\sqrt{6))}

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  3. Evaluate: sqrt(6+2\sqrt{3)+2sqrt(2)+2sqrt(6)}-(1)/(sqrt(5-2\sqrt{6))}

Expression: $\sqrt{ 6+2\sqrt{ 3 }+2\sqrt{ 2 }+2\sqrt{ 6 } }-\frac{ 1 }{ \sqrt{ 5-2\sqrt{ 6 } } }$

Use ${a}^{2}-2ab+{b}^{2}={\left( a-b \right)}^{2}$ to factor the expression

$\sqrt{ 6+2\sqrt{ 3 }+2\sqrt{ 2 }+2\sqrt{ 6 } }-\frac{ 1 }{ \sqrt{ {\left( \sqrt{ 2 }-\sqrt{ 3 } \right)}^{2} } }$

Reduce the index of the radical and exponent with $2$

$\sqrt{ 6+2\sqrt{ 3 }+2\sqrt{ 2 }+2\sqrt{ 6 } }-\frac{ 1 }{ \sqrt{ 3 }-\sqrt{ 2 } }$

Rationalize the denominator

$\sqrt{ 6+2\sqrt{ 3 }+2\sqrt{ 2 }+2\sqrt{ 6 } }-\left( \sqrt{ 3 }+\sqrt{ 2 } \right)$

When there is a $-$ in front of an expression in parentheses, change the sign of each term of the expression and remove the parentheses

$\begin{align*}&\sqrt{ 6+2\sqrt{ 3 }+2\sqrt{ 2 }+2\sqrt{ 6 } }-\sqrt{ 3 }-\sqrt{ 2 } \\&\approx1\end{align*}$

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