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Calculate the value
$\dfrac{ 1 }{ \sqrt{ 6 } + \sqrt{ 2 } } - \dfrac{ \sqrt{ 3 } }{ \sqrt{ 6 } - \sqrt{ 2 } }$
$- \sqrt{ 2 }$
Calculate the value
$\dfrac { 1 } { \sqrt{ 6 } + \sqrt{ 2 } } - \dfrac { \sqrt{ 3 } } { \sqrt{ 6 } - \sqrt{ 2 } }$
 Find the conjugate irrational number of denominator 
$\color{#FF6800}{ \dfrac { 1 } { \sqrt{ 6 } + \sqrt{ 2 } } } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { \sqrt{ 6 } - \sqrt{ 2 } } { \sqrt{ 6 } - \sqrt{ 2 } } } - \dfrac { \sqrt{ 3 } } { \sqrt{ 6 } - \sqrt{ 2 } }$
$\dfrac { 1 } { \sqrt{ 6 } + \sqrt{ 2 } } \times \dfrac { \sqrt{ 6 } - \sqrt{ 2 } } { \sqrt{ 6 } - \sqrt{ 2 } } - \dfrac { \sqrt{ 3 } } { \sqrt{ 6 } - \sqrt{ 2 } }$
 The denominator is multiplied by denominator, and the numerator is multiplied by numerator 
$\color{#FF6800}{ \dfrac { 1 \left ( \sqrt{ 6 } - \sqrt{ 2 } \right ) } { \left ( \sqrt{ 6 } + \sqrt{ 2 } \right ) \left ( \sqrt{ 6 } - \sqrt{ 2 } \right ) } } - \dfrac { \sqrt{ 3 } } { \sqrt{ 6 } - \sqrt{ 2 } }$
$\dfrac { \color{#FF6800}{ 1 } \left ( \sqrt{ \color{#FF6800}{ 6 } } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } \right ) } { \left ( \sqrt{ 6 } + \sqrt{ 2 } \right ) \left ( \sqrt{ 6 } - \sqrt{ 2 } \right ) } - \dfrac { \sqrt{ 3 } } { \sqrt{ 6 } - \sqrt{ 2 } }$
 Multiply each term in parentheses by $1$
$\dfrac { \color{#FF6800}{ 1 } \sqrt{ \color{#FF6800}{ 6 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } \right ) } { \left ( \sqrt{ 6 } + \sqrt{ 2 } \right ) \left ( \sqrt{ 6 } - \sqrt{ 2 } \right ) } - \dfrac { \sqrt{ 3 } } { \sqrt{ 6 } - \sqrt{ 2 } }$
$\dfrac { 1 \sqrt{ 6 } + 1 \times \left ( - \sqrt{ 2 } \right ) } { \left ( \sqrt{ \color{#FF6800}{ 6 } } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 2 } } \right ) \left ( \sqrt{ \color{#FF6800}{ 6 } } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } \right ) } - \dfrac { \sqrt{ 3 } } { \sqrt{ 6 } - \sqrt{ 2 } }$
 Expand the expression using $\left(a - b\right)\left(a + b\right) = a^{2} - b^{2}$
$\dfrac { 1 \sqrt{ 6 } + 1 \times \left ( - \sqrt{ 2 } \right ) } { \left ( \sqrt{ \color{#FF6800}{ 6 } } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \left ( \sqrt{ \color{#FF6800}{ 2 } } \right ) ^ { \color{#FF6800}{ 2 } } } - \dfrac { \sqrt{ 3 } } { \sqrt{ 6 } - \sqrt{ 2 } }$
$\dfrac { 1 \sqrt{ 6 } + 1 \times \left ( - \sqrt{ 2 } \right ) } { \left ( \sqrt{ \color{#FF6800}{ 6 } } \right ) ^ { \color{#FF6800}{ 2 } } - \left ( \sqrt{ 2 } \right ) ^ { 2 } } - \dfrac { \sqrt{ 3 } } { \sqrt{ 6 } - \sqrt{ 2 } }$
 Calculate power 
$\dfrac { 1 \sqrt{ 6 } + 1 \times \left ( - \sqrt{ 2 } \right ) } { \color{#FF6800}{ 6 } - \left ( \sqrt{ 2 } \right ) ^ { 2 } } - \dfrac { \sqrt{ 3 } } { \sqrt{ 6 } - \sqrt{ 2 } }$
$\dfrac { 1 \sqrt{ 6 } + 1 \times \left ( - \sqrt{ 2 } \right ) } { 6 - \left ( \sqrt{ \color{#FF6800}{ 2 } } \right ) ^ { \color{#FF6800}{ 2 } } } - \dfrac { \sqrt{ 3 } } { \sqrt{ 6 } - \sqrt{ 2 } }$
 Calculate power 
$\dfrac { 1 \sqrt{ 6 } + 1 \times \left ( - \sqrt{ 2 } \right ) } { 6 - \color{#FF6800}{ 2 } } - \dfrac { \sqrt{ 3 } } { \sqrt{ 6 } - \sqrt{ 2 } }$
$\dfrac { \color{#FF6800}{ 1 } \sqrt{ 6 } + 1 \times \left ( - \sqrt{ 2 } \right ) } { 6 - 2 } - \dfrac { \sqrt{ 3 } } { \sqrt{ 6 } - \sqrt{ 2 } }$
 Multiplying any number by 1 does not change the value 
$\dfrac { \sqrt{ 6 } + 1 \times \left ( - \sqrt{ 2 } \right ) } { 6 - 2 } - \dfrac { \sqrt{ 3 } } { \sqrt{ 6 } - \sqrt{ 2 } }$
$\dfrac { \sqrt{ 6 } + \color{#FF6800}{ 1 } \times \left ( - \sqrt{ 2 } \right ) } { 6 - 2 } - \dfrac { \sqrt{ 3 } } { \sqrt{ 6 } - \sqrt{ 2 } }$
 Multiplying any number by 1 does not change the value 
$\dfrac { \sqrt{ 6 } - \sqrt{ 2 } } { 6 - 2 } - \dfrac { \sqrt{ 3 } } { \sqrt{ 6 } - \sqrt{ 2 } }$
$\dfrac { \sqrt{ 6 } - \sqrt{ 2 } } { \color{#FF6800}{ 6 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } } - \dfrac { \sqrt{ 3 } } { \sqrt{ 6 } - \sqrt{ 2 } }$
 Subtract $2$ from $6$
$\dfrac { \sqrt{ 6 } - \sqrt{ 2 } } { \color{#FF6800}{ 4 } } - \dfrac { \sqrt{ 3 } } { \sqrt{ 6 } - \sqrt{ 2 } }$
$\dfrac { \sqrt{ 6 } - \sqrt{ 2 } } { 4 } - \dfrac { \sqrt{ 3 } } { \sqrt{ 6 } - \sqrt{ 2 } }$
 Find the conjugate irrational number of denominator 
$\dfrac { \sqrt{ 6 } - \sqrt{ 2 } } { 4 } - \left ( \color{#FF6800}{ \dfrac { \sqrt{ 3 } } { \sqrt{ 6 } - \sqrt{ 2 } } } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { \sqrt{ 6 } + \sqrt{ 2 } } { \sqrt{ 6 } + \sqrt{ 2 } } } \right )$
$\dfrac { \sqrt{ 6 } - \sqrt{ 2 } } { 4 } - \left ( \dfrac { \sqrt{ 3 } } { \sqrt{ 6 } - \sqrt{ 2 } } \times \dfrac { \sqrt{ 6 } + \sqrt{ 2 } } { \sqrt{ 6 } + \sqrt{ 2 } } \right )$
 The denominator is multiplied by denominator, and the numerator is multiplied by numerator 
$\dfrac { \sqrt{ 6 } - \sqrt{ 2 } } { 4 } - \color{#FF6800}{ \dfrac { \sqrt{ 3 } \left ( \sqrt{ 6 } + \sqrt{ 2 } \right ) } { \left ( \sqrt{ 6 } - \sqrt{ 2 } \right ) \left ( \sqrt{ 6 } + \sqrt{ 2 } \right ) } }$
$\dfrac { \sqrt{ 6 } - \sqrt{ 2 } } { 4 } - \dfrac { \sqrt{ \color{#FF6800}{ 3 } } \left ( \sqrt{ \color{#FF6800}{ 6 } } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 2 } } \right ) } { \left ( \sqrt{ 6 } - \sqrt{ 2 } \right ) \left ( \sqrt{ 6 } + \sqrt{ 2 } \right ) }$
 Multiply each term in parentheses by $\sqrt{ 3 }$
$\dfrac { \sqrt{ 6 } - \sqrt{ 2 } } { 4 } - \dfrac { \sqrt{ \color{#FF6800}{ 3 } } \sqrt{ \color{#FF6800}{ 6 } } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 3 } } \sqrt{ \color{#FF6800}{ 2 } } } { \left ( \sqrt{ 6 } - \sqrt{ 2 } \right ) \left ( \sqrt{ 6 } + \sqrt{ 2 } \right ) }$
$\dfrac { \sqrt{ 6 } - \sqrt{ 2 } } { 4 } - \dfrac { \sqrt{ 3 } \sqrt{ 6 } + \sqrt{ 3 } \sqrt{ 2 } } { \left ( \sqrt{ \color{#FF6800}{ 6 } } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } \right ) \left ( \sqrt{ \color{#FF6800}{ 6 } } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 2 } } \right ) }$
 Expand the expression using $\left(a - b\right)\left(a + b\right) = a^{2} - b^{2}$
$\dfrac { \sqrt{ 6 } - \sqrt{ 2 } } { 4 } - \dfrac { \sqrt{ 3 } \sqrt{ 6 } + \sqrt{ 3 } \sqrt{ 2 } } { \left ( \sqrt{ \color{#FF6800}{ 6 } } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \left ( \sqrt{ \color{#FF6800}{ 2 } } \right ) ^ { \color{#FF6800}{ 2 } } }$
$\dfrac { \sqrt{ 6 } - \sqrt{ 2 } } { 4 } - \dfrac { \sqrt{ \color{#FF6800}{ 3 } } \sqrt{ \color{#FF6800}{ 6 } } + \sqrt{ 3 } \sqrt{ 2 } } { \left ( \sqrt{ 6 } \right ) ^ { 2 } - \left ( \sqrt{ 2 } \right ) ^ { 2 } }$
 Arrange the expression 
$\dfrac { \sqrt{ 6 } - \sqrt{ 2 } } { 4 } - \dfrac { \sqrt{ \color{#FF6800}{ 3 } \color{#FF6800}{ \times } \color{#FF6800}{ 6 } } + \sqrt{ 3 } \sqrt{ 2 } } { \left ( \sqrt{ 6 } \right ) ^ { 2 } - \left ( \sqrt{ 2 } \right ) ^ { 2 } }$
$\dfrac { \sqrt{ 6 } - \sqrt{ 2 } } { 4 } - \dfrac { \sqrt{ 3 \times 6 } + \sqrt{ \color{#FF6800}{ 3 } } \sqrt{ \color{#FF6800}{ 2 } } } { \left ( \sqrt{ 6 } \right ) ^ { 2 } - \left ( \sqrt{ 2 } \right ) ^ { 2 } }$
 Arrange the expression 
$\dfrac { \sqrt{ 6 } - \sqrt{ 2 } } { 4 } - \dfrac { \sqrt{ 3 \times 6 } + \sqrt{ \color{#FF6800}{ 3 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } } } { \left ( \sqrt{ 6 } \right ) ^ { 2 } - \left ( \sqrt{ 2 } \right ) ^ { 2 } }$
$\dfrac { \sqrt{ 6 } - \sqrt{ 2 } } { 4 } - \dfrac { \sqrt{ 3 \times 6 } + \sqrt{ 3 \times 2 } } { \left ( \sqrt{ \color{#FF6800}{ 6 } } \right ) ^ { \color{#FF6800}{ 2 } } - \left ( \sqrt{ 2 } \right ) ^ { 2 } }$
 Calculate power 
$\dfrac { \sqrt{ 6 } - \sqrt{ 2 } } { 4 } - \dfrac { \sqrt{ 3 \times 6 } + \sqrt{ 3 \times 2 } } { \color{#FF6800}{ 6 } - \left ( \sqrt{ 2 } \right ) ^ { 2 } }$
$\dfrac { \sqrt{ 6 } - \sqrt{ 2 } } { 4 } - \dfrac { \sqrt{ 3 \times 6 } + \sqrt{ 3 \times 2 } } { 6 - \left ( \sqrt{ \color{#FF6800}{ 2 } } \right ) ^ { \color{#FF6800}{ 2 } } }$
 Calculate power 
$\dfrac { \sqrt{ 6 } - \sqrt{ 2 } } { 4 } - \dfrac { \sqrt{ 3 \times 6 } + \sqrt{ 3 \times 2 } } { 6 - \color{#FF6800}{ 2 } }$
$\dfrac { \sqrt{ 6 } - \sqrt{ 2 } } { 4 } - \dfrac { \sqrt{ \color{#FF6800}{ 3 } \color{#FF6800}{ \times } \color{#FF6800}{ 6 } } + \sqrt{ 3 \times 2 } } { 6 - 2 }$
 Multiply $3$ and $6$
$\dfrac { \sqrt{ 6 } - \sqrt{ 2 } } { 4 } - \dfrac { \sqrt{ \color{#FF6800}{ 18 } } + \sqrt{ 3 \times 2 } } { 6 - 2 }$
$\dfrac { \sqrt{ 6 } - \sqrt{ 2 } } { 4 } - \dfrac { \sqrt{ 18 } + \sqrt{ \color{#FF6800}{ 3 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } } } { 6 - 2 }$
 Multiply $3$ and $2$
$\dfrac { \sqrt{ 6 } - \sqrt{ 2 } } { 4 } - \dfrac { \sqrt{ 18 } + \sqrt{ \color{#FF6800}{ 6 } } } { 6 - 2 }$
$\dfrac { \sqrt{ 6 } - \sqrt{ 2 } } { 4 } - \dfrac { \sqrt{ 18 } + \sqrt{ 6 } } { \color{#FF6800}{ 6 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } }$
 Subtract $2$ from $6$
$\dfrac { \sqrt{ 6 } - \sqrt{ 2 } } { 4 } - \dfrac { \sqrt{ 18 } + \sqrt{ 6 } } { \color{#FF6800}{ 4 } }$
$\dfrac { \sqrt{ 6 } - \sqrt{ 2 } } { 4 } - \dfrac { \sqrt{ \color{#FF6800}{ 18 } } + \sqrt{ 6 } } { 4 }$
 Organize the part that can be taken out of the radical sign inside the square root symbol 
$\dfrac { \sqrt{ 6 } - \sqrt{ 2 } } { 4 } - \dfrac { \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 2 } } + \sqrt{ 6 } } { 4 }$
$\color{#FF6800}{ \dfrac { \sqrt{ 6 } - \sqrt{ 2 } } { 4 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 3 \sqrt{ 2 } + \sqrt{ 6 } } { 4 } }$
 Combine the fraction with the same denominator 
$\dfrac { \sqrt{ \color{#FF6800}{ 6 } } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 6 } } \right ) } { 4 }$
$\dfrac { \sqrt{ 6 } - \sqrt{ 2 } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 6 } } \right ) } { 4 }$
 Change the symbol of each term in parentheses when there is a (-) symbol in front of parentheses 
$\dfrac { \sqrt{ 6 } - \sqrt{ 2 } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 6 } } } { 4 }$
$\dfrac { \sqrt{ \color{#FF6800}{ 6 } } - \sqrt{ 2 } - 3 \sqrt{ 2 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 6 } } } { 4 }$
 Remove the two numbers if the values are the same and the signs are different 
$\dfrac { - \sqrt{ 2 } - 3 \sqrt{ 2 } } { 4 }$
$\dfrac { \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 2 } } } { 4 }$
 Calculate between similar terms 
$\dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 2 } } } { 4 }$
$\color{#FF6800}{ \dfrac { - 4 \sqrt{ 2 } } { 4 } }$
 Reduce the fraction 
$\color{#FF6800}{ - } \color{#FF6800}{ 1 } \sqrt{ \color{#FF6800}{ 2 } }$
$\color{#FF6800}{ - } \color{#FF6800}{ 1 } \sqrt{ 2 }$
 Multiplying any number by 1 does not change the value 
$- \sqrt{ 2 }$
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