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Calculate the value
$\dfrac{ \sqrt{ 7 } - \sqrt{ 3 } }{ \sqrt{ 7 } + \sqrt{ 3 } }$
$\dfrac { 5 - \sqrt{ 21 } } { 2 }$
Calculate the value
$\dfrac { \sqrt{ 7 } - \sqrt{ 3 } } { \sqrt{ 7 } + \sqrt{ 3 } }$
 Find the conjugate irrational number of denominator 
$\color{#FF6800}{ \dfrac { \sqrt{ 7 } - \sqrt{ 3 } } { \sqrt{ 7 } + \sqrt{ 3 } } } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { \sqrt{ 7 } - \sqrt{ 3 } } { \sqrt{ 7 } - \sqrt{ 3 } } }$
$\dfrac { \sqrt{ 7 } - \sqrt{ 3 } } { \sqrt{ 7 } + \sqrt{ 3 } } \times \dfrac { \sqrt{ 7 } - \sqrt{ 3 } } { \sqrt{ 7 } - \sqrt{ 3 } }$
 The denominator is multiplied by denominator, and the numerator is multiplied by numerator 
$\color{#FF6800}{ \dfrac { \left ( \sqrt{ 7 } - \sqrt{ 3 } \right ) \left ( \sqrt{ 7 } - \sqrt{ 3 } \right ) } { \left ( \sqrt{ 7 } + \sqrt{ 3 } \right ) \left ( \sqrt{ 7 } - \sqrt{ 3 } \right ) } }$
$\dfrac { \left ( \sqrt{ \color{#FF6800}{ 7 } } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 3 } } \right ) \left ( \sqrt{ \color{#FF6800}{ 7 } } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 3 } } \right ) } { \left ( \sqrt{ 7 } + \sqrt{ 3 } \right ) \left ( \sqrt{ 7 } - \sqrt{ 3 } \right ) }$
 Expand using $\left(a + b\right)\left(c + d\right) = ac + ad + bc + bd$
$\dfrac { \sqrt{ \color{#FF6800}{ 7 } } \sqrt{ \color{#FF6800}{ 7 } } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 7 } } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 3 } } \right ) \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 3 } } \sqrt{ \color{#FF6800}{ 7 } } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 3 } } \right ) } { \left ( \sqrt{ 7 } + \sqrt{ 3 } \right ) \left ( \sqrt{ 7 } - \sqrt{ 3 } \right ) }$
$\dfrac { \sqrt{ 7 } \sqrt{ 7 } + \sqrt{ 7 } \times \left ( - \sqrt{ 3 } \right ) - \sqrt{ 3 } \sqrt{ 7 } - \sqrt{ 3 } \times \left ( - \sqrt{ 3 } \right ) } { \left ( \sqrt{ \color{#FF6800}{ 7 } } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 3 } } \right ) \left ( \sqrt{ \color{#FF6800}{ 7 } } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 3 } } \right ) }$
 Expand the expression using $\left(a - b\right)\left(a + b\right) = a^{2} - b^{2}$
$\dfrac { \sqrt{ 7 } \sqrt{ 7 } + \sqrt{ 7 } \times \left ( - \sqrt{ 3 } \right ) - \sqrt{ 3 } \sqrt{ 7 } - \sqrt{ 3 } \times \left ( - \sqrt{ 3 } \right ) } { \left ( \sqrt{ \color{#FF6800}{ 7 } } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \left ( \sqrt{ \color{#FF6800}{ 3 } } \right ) ^ { \color{#FF6800}{ 2 } } }$
$\dfrac { \sqrt{ \color{#FF6800}{ 7 } } \sqrt{ \color{#FF6800}{ 7 } } + \sqrt{ 7 } \times \left ( - \sqrt{ 3 } \right ) - \sqrt{ 3 } \sqrt{ 7 } - \sqrt{ 3 } \times \left ( - \sqrt{ 3 } \right ) } { \left ( \sqrt{ 7 } \right ) ^ { 2 } - \left ( \sqrt{ 3 } \right ) ^ { 2 } }$
 Arrange the expression 
$\dfrac { \sqrt{ \color{#FF6800}{ 7 } \color{#FF6800}{ \times } \color{#FF6800}{ 7 } } + \sqrt{ 7 } \times \left ( - \sqrt{ 3 } \right ) - \sqrt{ 3 } \sqrt{ 7 } - \sqrt{ 3 } \times \left ( - \sqrt{ 3 } \right ) } { \left ( \sqrt{ 7 } \right ) ^ { 2 } - \left ( \sqrt{ 3 } \right ) ^ { 2 } }$
$\dfrac { \sqrt{ 7 \times 7 } + \sqrt{ 7 } \times \left ( - \sqrt{ 3 } \right ) - \sqrt{ 3 } \sqrt{ 7 } - \sqrt{ 3 } \times \left ( - \sqrt{ 3 } \right ) } { \left ( \sqrt{ \color{#FF6800}{ 7 } } \right ) ^ { \color{#FF6800}{ 2 } } - \left ( \sqrt{ 3 } \right ) ^ { 2 } }$
 Calculate power 
$\dfrac { \sqrt{ 7 \times 7 } + \sqrt{ 7 } \times \left ( - \sqrt{ 3 } \right ) - \sqrt{ 3 } \sqrt{ 7 } - \sqrt{ 3 } \times \left ( - \sqrt{ 3 } \right ) } { \color{#FF6800}{ 7 } - \left ( \sqrt{ 3 } \right ) ^ { 2 } }$
$\dfrac { \sqrt{ 7 \times 7 } + \sqrt{ 7 } \times \left ( - \sqrt{ 3 } \right ) - \sqrt{ 3 } \sqrt{ 7 } - \sqrt{ 3 } \times \left ( - \sqrt{ 3 } \right ) } { 7 - \left ( \sqrt{ \color{#FF6800}{ 3 } } \right ) ^ { \color{#FF6800}{ 2 } } }$
 Calculate power 
$\dfrac { \sqrt{ 7 \times 7 } + \sqrt{ 7 } \times \left ( - \sqrt{ 3 } \right ) - \sqrt{ 3 } \sqrt{ 7 } - \sqrt{ 3 } \times \left ( - \sqrt{ 3 } \right ) } { 7 - \color{#FF6800}{ 3 } }$
$\dfrac { \sqrt{ \color{#FF6800}{ 7 } \color{#FF6800}{ \times } \color{#FF6800}{ 7 } } + \sqrt{ 7 } \times \left ( - \sqrt{ 3 } \right ) - \sqrt{ 3 } \sqrt{ 7 } - \sqrt{ 3 } \times \left ( - \sqrt{ 3 } \right ) } { 7 - 3 }$
 Multiply $7$ and $7$
$\dfrac { \sqrt{ \color{#FF6800}{ 49 } } + \sqrt{ 7 } \times \left ( - \sqrt{ 3 } \right ) - \sqrt{ 3 } \sqrt{ 7 } - \sqrt{ 3 } \times \left ( - \sqrt{ 3 } \right ) } { 7 - 3 }$
$\dfrac { \sqrt{ 49 } + \sqrt{ 7 } \times \left ( \color{#FF6800}{ - } \sqrt{ 3 } \right ) - \sqrt{ 3 } \sqrt{ 7 } - \sqrt{ 3 } \times \left ( - \sqrt{ 3 } \right ) } { 7 - 3 }$
 Move the (-) sign forward 
$\dfrac { \sqrt{ 49 } \color{#FF6800}{ - } \sqrt{ 7 } \sqrt{ 3 } - \sqrt{ 3 } \sqrt{ 7 } - \sqrt{ 3 } \times \left ( - \sqrt{ 3 } \right ) } { 7 - 3 }$
$\dfrac { \sqrt{ 49 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 7 } } \sqrt{ \color{#FF6800}{ 3 } } - \sqrt{ 3 } \sqrt{ 7 } - \sqrt{ 3 } \times \left ( - \sqrt{ 3 } \right ) } { 7 - 3 }$
 Calculate multiplication 
$\dfrac { \sqrt{ 49 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 21 } } - \sqrt{ 3 } \sqrt{ 7 } - \sqrt{ 3 } \times \left ( - \sqrt{ 3 } \right ) } { 7 - 3 }$
$\dfrac { \sqrt{ 49 } - \sqrt{ 21 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 3 } } \sqrt{ \color{#FF6800}{ 7 } } - \sqrt{ 3 } \times \left ( - \sqrt{ 3 } \right ) } { 7 - 3 }$
 Calculate multiplication 
$\dfrac { \sqrt{ 49 } - \sqrt{ 21 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 21 } } - \sqrt{ 3 } \times \left ( - \sqrt{ 3 } \right ) } { 7 - 3 }$
$\dfrac { \sqrt{ 49 } - \sqrt{ 21 } - \sqrt{ 21 } \color{#FF6800}{ - } \sqrt{ 3 } \times \left ( \color{#FF6800}{ - } \sqrt{ 3 } \right ) } { 7 - 3 }$
 Since negative numbers are multiplied by an even number, remove the (-) sign 
$\dfrac { \sqrt{ 49 } - \sqrt{ 21 } - \sqrt{ 21 } + \sqrt{ 3 } \sqrt{ 3 } } { 7 - 3 }$
$\dfrac { \sqrt{ 49 } - \sqrt{ 21 } - \sqrt{ 21 } + \sqrt{ \color{#FF6800}{ 3 } } \sqrt{ 3 } } { 7 - 3 }$
 If the exponent is omitted, the exponent of that term is equal to 1 
$\dfrac { \sqrt{ 49 } - \sqrt{ 21 } - \sqrt{ 21 } + \left ( \sqrt{ \color{#FF6800}{ 3 } } \right ) ^ { \color{#FF6800}{ 1 } } \sqrt{ 3 } } { 7 - 3 }$
$\dfrac { \sqrt{ 49 } - \sqrt{ 21 } - \sqrt{ 21 } + \left ( \sqrt{ 3 } \right ) ^ { 1 } \sqrt{ \color{#FF6800}{ 3 } } } { 7 - 3 }$
 If the exponent is omitted, the exponent of that term is equal to 1 
$\dfrac { \sqrt{ 49 } - \sqrt{ 21 } - \sqrt{ 21 } + \left ( \sqrt{ 3 } \right ) ^ { 1 } \left ( \sqrt{ \color{#FF6800}{ 3 } } \right ) ^ { \color{#FF6800}{ 1 } } } { 7 - 3 }$
$\dfrac { \sqrt{ 49 } - \sqrt{ 21 } - \sqrt{ 21 } + \left ( \sqrt{ \color{#FF6800}{ 3 } } \right ) ^ { \color{#FF6800}{ 1 } } \left ( \sqrt{ \color{#FF6800}{ 3 } } \right ) ^ { \color{#FF6800}{ 1 } } } { 7 - 3 }$
 Add the exponent as the base is the same 
$\dfrac { \sqrt{ 49 } - \sqrt{ 21 } - \sqrt{ 21 } + \left ( \sqrt{ \color{#FF6800}{ 3 } } \right ) ^ { \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } } } { 7 - 3 }$
$\dfrac { \sqrt{ 49 } - \sqrt{ 21 } - \sqrt{ 21 } + \left ( \sqrt{ 3 } \right ) ^ { \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } } } { 7 - 3 }$
 Add $1$ and $1$
$\dfrac { \sqrt{ 49 } - \sqrt{ 21 } - \sqrt{ 21 } + \left ( \sqrt{ 3 } \right ) ^ { \color{#FF6800}{ 2 } } } { 7 - 3 }$
$\dfrac { \sqrt{ 49 } - \sqrt{ 21 } - \sqrt{ 21 } + \left ( \sqrt{ 3 } \right ) ^ { 2 } } { \color{#FF6800}{ 7 } \color{#FF6800}{ - } \color{#FF6800}{ 3 } }$
 Subtract $3$ from $7$
$\dfrac { \sqrt{ 49 } - \sqrt{ 21 } - \sqrt{ 21 } + \left ( \sqrt{ 3 } \right ) ^ { 2 } } { \color{#FF6800}{ 4 } }$
$\dfrac { \sqrt{ \color{#FF6800}{ 49 } } - \sqrt{ 21 } - \sqrt{ 21 } + \left ( \sqrt{ 3 } \right ) ^ { 2 } } { 4 }$
 Organize the part that can be taken out of the radical sign inside the square root symbol 
$\dfrac { \color{#FF6800}{ 7 } - \sqrt{ 21 } - \sqrt{ 21 } + \left ( \sqrt{ 3 } \right ) ^ { 2 } } { 4 }$
$\dfrac { 7 - \sqrt{ 21 } - \sqrt{ 21 } + \left ( \sqrt{ \color{#FF6800}{ 3 } } \right ) ^ { \color{#FF6800}{ 2 } } } { 4 }$
 If you square the radical sign, it will disappear 
$\dfrac { 7 - \sqrt{ 21 } - \sqrt{ 21 } + \color{#FF6800}{ 3 } } { 4 }$
$\dfrac { \color{#FF6800}{ 7 } - \sqrt{ 21 } - \sqrt{ 21 } \color{#FF6800}{ + } \color{#FF6800}{ 3 } } { 4 }$
 Add $7$ and $3$
$\dfrac { \color{#FF6800}{ 10 } - \sqrt{ 21 } - \sqrt{ 21 } } { 4 }$
$\dfrac { 10 \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 21 } } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 21 } } } { 4 }$
 Calculate between similar terms 
$\dfrac { 10 \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 21 } } } { 4 }$
$\color{#FF6800}{ \dfrac { 10 - 2 \sqrt{ 21 } } { 4 } }$
 Reduce the fraction 
$\color{#FF6800}{ \dfrac { 5 - \sqrt{ 21 } } { 2 } }$
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