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Formula
Calculate the value
$\dfrac{ \sqrt{ 6 } }{ \sqrt{ 2 } + \sqrt{ 3 } }$
$- 2 \sqrt{ 3 } + 3 \sqrt{ 2 }$
Calculate the value
$\dfrac { \sqrt{ 6 } } { \sqrt{ 2 } + \sqrt{ 3 } }$
 Find the conjugate irrational number of denominator 
$\color{#FF6800}{ \dfrac { \sqrt{ 6 } } { \sqrt{ 2 } + \sqrt{ 3 } } } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { \sqrt{ 2 } - \sqrt{ 3 } } { \sqrt{ 2 } - \sqrt{ 3 } } }$
$\dfrac { \sqrt{ 6 } } { \sqrt{ 2 } + \sqrt{ 3 } } \times \dfrac { \sqrt{ 2 } - \sqrt{ 3 } } { \sqrt{ 2 } - \sqrt{ 3 } }$
 The denominator is multiplied by denominator, and the numerator is multiplied by numerator 
$\color{#FF6800}{ \dfrac { \sqrt{ 6 } \left ( \sqrt{ 2 } - \sqrt{ 3 } \right ) } { \left ( \sqrt{ 2 } + \sqrt{ 3 } \right ) \left ( \sqrt{ 2 } - \sqrt{ 3 } \right ) } }$
$\dfrac { \sqrt{ \color{#FF6800}{ 6 } } \left ( \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 3 } } \right ) } { \left ( \sqrt{ 2 } + \sqrt{ 3 } \right ) \left ( \sqrt{ 2 } - \sqrt{ 3 } \right ) }$
 Multiply each term in parentheses by $\sqrt{ 6 }$
$\dfrac { \sqrt{ \color{#FF6800}{ 6 } } \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 6 } } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 3 } } \right ) } { \left ( \sqrt{ 2 } + \sqrt{ 3 } \right ) \left ( \sqrt{ 2 } - \sqrt{ 3 } \right ) }$
$\dfrac { \sqrt{ 6 } \sqrt{ 2 } + \sqrt{ 6 } \times \left ( - \sqrt{ 3 } \right ) } { \left ( \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 3 } } \right ) \left ( \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 3 } } \right ) }$
 Expand the expression using $\left(a - b\right)\left(a + b\right) = a^{2} - b^{2}$
$\dfrac { \sqrt{ 6 } \sqrt{ 2 } + \sqrt{ 6 } \times \left ( - \sqrt{ 3 } \right ) } { \left ( \sqrt{ \color{#FF6800}{ 2 } } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \left ( \sqrt{ \color{#FF6800}{ 3 } } \right ) ^ { \color{#FF6800}{ 2 } } }$
$\dfrac { \sqrt{ \color{#FF6800}{ 6 } } \sqrt{ \color{#FF6800}{ 2 } } + \sqrt{ 6 } \times \left ( - \sqrt{ 3 } \right ) } { \left ( \sqrt{ 2 } \right ) ^ { 2 } - \left ( \sqrt{ 3 } \right ) ^ { 2 } }$
 Arrange the expression 
$\dfrac { \sqrt{ \color{#FF6800}{ 6 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } } + \sqrt{ 6 } \times \left ( - \sqrt{ 3 } \right ) } { \left ( \sqrt{ 2 } \right ) ^ { 2 } - \left ( \sqrt{ 3 } \right ) ^ { 2 } }$
$\dfrac { \sqrt{ 6 \times 2 } + \sqrt{ 6 } \times \left ( - \sqrt{ 3 } \right ) } { \left ( \sqrt{ \color{#FF6800}{ 2 } } \right ) ^ { \color{#FF6800}{ 2 } } - \left ( \sqrt{ 3 } \right ) ^ { 2 } }$
 Calculate power 
$\dfrac { \sqrt{ 6 \times 2 } + \sqrt{ 6 } \times \left ( - \sqrt{ 3 } \right ) } { \color{#FF6800}{ 2 } - \left ( \sqrt{ 3 } \right ) ^ { 2 } }$
$\dfrac { \sqrt{ 6 \times 2 } + \sqrt{ 6 } \times \left ( - \sqrt{ 3 } \right ) } { 2 - \left ( \sqrt{ \color{#FF6800}{ 3 } } \right ) ^ { \color{#FF6800}{ 2 } } }$
 Calculate power 
$\dfrac { \sqrt{ 6 \times 2 } + \sqrt{ 6 } \times \left ( - \sqrt{ 3 } \right ) } { 2 - \color{#FF6800}{ 3 } }$
$\dfrac { \sqrt{ \color{#FF6800}{ 6 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } } + \sqrt{ 6 } \times \left ( - \sqrt{ 3 } \right ) } { 2 - 3 }$
 Multiply $6$ and $2$
$\dfrac { \sqrt{ \color{#FF6800}{ 12 } } + \sqrt{ 6 } \times \left ( - \sqrt{ 3 } \right ) } { 2 - 3 }$
$\dfrac { \sqrt{ 12 } + \sqrt{ 6 } \times \left ( \color{#FF6800}{ - } \sqrt{ 3 } \right ) } { 2 - 3 }$
 Move the (-) sign forward 
$\dfrac { \sqrt{ 12 } \color{#FF6800}{ - } \sqrt{ 6 } \sqrt{ 3 } } { 2 - 3 }$
$\dfrac { \sqrt{ 12 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 6 } } \sqrt{ \color{#FF6800}{ 3 } } } { 2 - 3 }$
 Calculate multiplication 
$\dfrac { \sqrt{ 12 } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 2 } } \right ) } { 2 - 3 }$
$\dfrac { \sqrt{ 12 } - \left ( 3 \sqrt{ 2 } \right ) } { \color{#FF6800}{ 2 } \color{#FF6800}{ - } \color{#FF6800}{ 3 } }$
 Subtract $3$ from $2$
$\dfrac { \sqrt{ 12 } - \left ( 3 \sqrt{ 2 } \right ) } { \color{#FF6800}{ - } \color{#FF6800}{ 1 } }$
$\dfrac { \sqrt{ 12 } - \left ( 3 \sqrt{ 2 } \right ) } { \color{#FF6800}{ - } 1 }$
 If the denominator is 1, the denominator can be removed 
$\color{#FF6800}{ - } \left ( \sqrt{ \color{#FF6800}{ 12 } } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 2 } } \right ) \right )$
$- \left ( \sqrt{ \color{#FF6800}{ 12 } } - \left ( 3 \sqrt{ 2 } \right ) \right )$
 Organize the part that can be taken out of the radical sign inside the square root symbol 
$- \left ( \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 3 } } - \left ( 3 \sqrt{ 2 } \right ) \right )$
$- \left ( 2 \sqrt{ 3 } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 2 } } \right ) \right )$
 Get rid of unnecessary parentheses 
$- \left ( 2 \sqrt{ 3 } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 2 } } \right )$
$\color{#FF6800}{ - } \left ( \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 2 } } \right )$
 Change the symbol of each term in parentheses when there is a (-) symbol in front of parentheses 
$\color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 2 } }$
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