Question

1 by 1- root 2 +root 3​

Answer 1

The given expression is,

\frac { 1 } { ( 1 + \sqrt { 2 } - \sqrt { 3 } ) }(1+2−3)1

Rationalising the denominator

\begin{gathered}\begin{array} { c } { \frac { 1 } { ( 1 + \sqrt { 2 } ) - \sqrt { 3 } } \times \frac { ( ( 1 + \sqrt { 2 } ) + \sqrt { 3 } ) } { ( 1 + \sqrt { 2 } ) + \sqrt { 3 } } } \\\\ { = \frac { ( ( 1 + \sqrt { 2 } ) + \sqrt { 3 } ) } { \left( ( 1 + \sqrt { 2 } ) ^ { 2 } - ( \sqrt { 3 } ) ^ { 2 } \right) } } \end{array}\end{gathered}(1+2)−31×(1+2)+3((1+2)+3)=((1+2)2−(3)2)((1+2)+3)

(As (a^2-b^2) = (a+b)(a-b)(a2−b2)=(a+b)(a−b) )

\begin{gathered}\begin{aligned} = & \frac { ( ( 1 + \sqrt { 2 } ) + \sqrt { 3 } ) } { ( ( 1 + 2 + 2 \sqrt { 2 } ) - 3 ) } \\\\ = & \frac { ( ( 1 + \sqrt { 2 } ) + \sqrt { 3 } ) } { 3 + 2 \sqrt { 2 } - 3 } \\\\ & = \frac { ( ( 1 + \sqrt { 2 } ) + \sqrt { 3 } ) } { 2 \sqrt { 2 } } \end{aligned}\end{gathered}==((1+2+22)−3)((1+2)+3)3+22−3((1+2)+3)=22((1+2)+3)

Multiply and divide by \sqrt{2}2

= \frac { ( ( 1 + \sqrt { 2 } ) + \sqrt { 3 } ) } { 2 \sqrt { 2 } } \times \frac { \sqrt { 2 } } { \sqrt { 2 } }=22((1+2)+3)×22

After rationalising the given denominator, we get the answer as

= \frac { \sqrt { 2 } + 2 + \sqrt { 6 } } { 4 }=42+2+6