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$\dfrac{ \sqrt{ 7 } - \sqrt{ 3 } }{ \sqrt{ 7 } + \sqrt{ 3 } }$
$\dfrac { 5 - \sqrt{ 21 } } { 2 }$
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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