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$9 - 4 \sqrt{ 5 }$
Calculate the value
$\dfrac { \sqrt{ 10 } - \sqrt{ \color{#FF6800}{ 8 } } } { \sqrt{ 10 } + \sqrt{ 8 } }$
$ $ Organize the part that can be taken out of the radical sign inside the square root symbol $ $
$\dfrac { \sqrt{ 10 } - \left ( \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \right ) } { \sqrt{ 10 } + \sqrt{ 8 } }$
$\dfrac { \sqrt{ 10 } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \right ) } { \sqrt{ 10 } + \sqrt{ 8 } }$
$ $ Get rid of unnecessary parentheses $ $
$\dfrac { \sqrt{ 10 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } } { \sqrt{ 10 } + \sqrt{ 8 } }$
$\dfrac { \sqrt{ 10 } - 2 \sqrt{ 2 } } { \sqrt{ 10 } + \sqrt{ \color{#FF6800}{ 8 } } }$
$ $ Organize the part that can be taken out of the radical sign inside the square root symbol $ $
$\dfrac { \sqrt{ 10 } - 2 \sqrt{ 2 } } { \sqrt{ 10 } + \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } }$
$\dfrac { \sqrt{ 10 } - 2 \sqrt{ 2 } } { \sqrt{ 10 } + 2 \sqrt{ 2 } }$
$ $ Find the conjugate irrational number of denominator $ $
$\color{#FF6800}{ \dfrac { \sqrt{ \color{#FF6800}{ 10 } } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } } { \sqrt{ \color{#FF6800}{ 10 } } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } } } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { \sqrt{ \color{#FF6800}{ 10 } } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \right ) } { \sqrt{ \color{#FF6800}{ 10 } } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \right ) } }$
$\dfrac { \sqrt{ 10 } - 2 \sqrt{ 2 } } { \sqrt{ 10 } + 2 \sqrt{ 2 } } \times \dfrac { \sqrt{ 10 } - \left ( 2 \sqrt{ 2 } \right ) } { \sqrt{ 10 } - \left ( 2 \sqrt{ 2 } \right ) }$
$ $ The denominator is multiplied by denominator, and the numerator is multiplied by numerator $ $
$\color{#FF6800}{ \dfrac { \left ( \sqrt{ \color{#FF6800}{ 10 } } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \right ) \left ( \sqrt{ \color{#FF6800}{ 10 } } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \right ) \right ) } { \left ( \sqrt{ \color{#FF6800}{ 10 } } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \right ) \left ( \sqrt{ \color{#FF6800}{ 10 } } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \right ) \right ) } }$
$\dfrac { \left ( \sqrt{ \color{#FF6800}{ 10 } } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \right ) \left ( \sqrt{ \color{#FF6800}{ 10 } } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \right ) \right ) } { \left ( \sqrt{ 10 } + 2 \sqrt{ 2 } \right ) \left ( \sqrt{ 10 } - \left ( 2 \sqrt{ 2 } \right ) \right ) }$
$ $ Expand using $ \left(a + b\right)\left(c + d\right) = ac + ad + bc + bd$
$\dfrac { \sqrt{ \color{#FF6800}{ 10 } } \sqrt{ \color{#FF6800}{ 10 } } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 10 } } \left ( \color{#FF6800}{ - } \left ( \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \right ) \right ) \color{#FF6800}{ + } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \right ) \sqrt{ \color{#FF6800}{ 10 } } \color{#FF6800}{ + } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \right ) \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \left ( \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \right ) \right ) } { \left ( \sqrt{ 10 } + 2 \sqrt{ 2 } \right ) \left ( \sqrt{ 10 } - \left ( 2 \sqrt{ 2 } \right ) \right ) }$
$\dfrac { \sqrt{ 10 } \sqrt{ 10 } + \sqrt{ 10 } \left ( - \left ( 2 \sqrt{ 2 } \right ) \right ) + \left ( - 2 \sqrt{ 2 } \right ) \sqrt{ 10 } + \left ( - 2 \sqrt{ 2 } \right ) \times \left ( - \left ( 2 \sqrt{ 2 } \right ) \right ) } { \left ( \sqrt{ \color{#FF6800}{ 10 } } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \right ) \left ( \sqrt{ \color{#FF6800}{ 10 } } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \right ) \right ) }$
$ $ Expand the expression using $ \left(a - b\right)\left(a + b\right) = a^{2} - b^{2}$
$\dfrac { \sqrt{ 10 } \sqrt{ 10 } + \sqrt{ 10 } \left ( - \left ( 2 \sqrt{ 2 } \right ) \right ) + \left ( - 2 \sqrt{ 2 } \right ) \sqrt{ 10 } + \left ( - 2 \sqrt{ 2 } \right ) \times \left ( - \left ( 2 \sqrt{ 2 } \right ) \right ) } { \left ( \sqrt{ \color{#FF6800}{ 10 } } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \right ) ^ { \color{#FF6800}{ 2 } } }$
$\dfrac { \sqrt{ \color{#FF6800}{ 10 } } \sqrt{ \color{#FF6800}{ 10 } } + \sqrt{ 10 } \left ( - \left ( 2 \sqrt{ 2 } \right ) \right ) + \left ( - 2 \sqrt{ 2 } \right ) \sqrt{ 10 } + \left ( - 2 \sqrt{ 2 } \right ) \times \left ( - \left ( 2 \sqrt{ 2 } \right ) \right ) } { \left ( \sqrt{ 10 } \right ) ^ { 2 } - \left ( 2 \sqrt{ 2 } \right ) ^ { 2 } }$
$ $ Arrange the expression $ $
$\dfrac { \sqrt{ \color{#FF6800}{ 10 } \color{#FF6800}{ \times } \color{#FF6800}{ 10 } } + \sqrt{ 10 } \left ( - \left ( 2 \sqrt{ 2 } \right ) \right ) + \left ( - 2 \sqrt{ 2 } \right ) \sqrt{ 10 } + \left ( - 2 \sqrt{ 2 } \right ) \times \left ( - \left ( 2 \sqrt{ 2 } \right ) \right ) } { \left ( \sqrt{ 10 } \right ) ^ { 2 } - \left ( 2 \sqrt{ 2 } \right ) ^ { 2 } }$
$\dfrac { \sqrt{ 10 \times 10 } + \sqrt{ 10 } \left ( - \left ( 2 \sqrt{ 2 } \right ) \right ) + \left ( - 2 \sqrt{ 2 } \right ) \sqrt{ 10 } + \left ( - 2 \sqrt{ 2 } \right ) \times \left ( - \left ( 2 \sqrt{ 2 } \right ) \right ) } { \left ( \sqrt{ \color{#FF6800}{ 10 } } \right ) ^ { \color{#FF6800}{ 2 } } - \left ( 2 \sqrt{ 2 } \right ) ^ { 2 } }$
$ $ Calculate power $ $
$\dfrac { \sqrt{ 10 \times 10 } + \sqrt{ 10 } \left ( - \left ( 2 \sqrt{ 2 } \right ) \right ) + \left ( - 2 \sqrt{ 2 } \right ) \sqrt{ 10 } + \left ( - 2 \sqrt{ 2 } \right ) \times \left ( - \left ( 2 \sqrt{ 2 } \right ) \right ) } { \color{#FF6800}{ 10 } - \left ( 2 \sqrt{ 2 } \right ) ^ { 2 } }$
$\dfrac { \sqrt{ 10 \times 10 } + \sqrt{ 10 } \left ( - \left ( 2 \sqrt{ 2 } \right ) \right ) + \left ( - 2 \sqrt{ 2 } \right ) \sqrt{ 10 } + \left ( - 2 \sqrt{ 2 } \right ) \times \left ( - \left ( 2 \sqrt{ 2 } \right ) \right ) } { 10 - \left ( \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \right ) ^ { \color{#FF6800}{ 2 } } }$
$ $ Calculate power $ $
$\dfrac { \sqrt{ 10 \times 10 } + \sqrt{ 10 } \left ( - \left ( 2 \sqrt{ 2 } \right ) \right ) + \left ( - 2 \sqrt{ 2 } \right ) \sqrt{ 10 } + \left ( - 2 \sqrt{ 2 } \right ) \times \left ( - \left ( 2 \sqrt{ 2 } \right ) \right ) } { 10 - \color{#FF6800}{ 8 } }$
$\dfrac { \sqrt{ \color{#FF6800}{ 10 } \color{#FF6800}{ \times } \color{#FF6800}{ 10 } } + \sqrt{ 10 } \left ( - \left ( 2 \sqrt{ 2 } \right ) \right ) + \left ( - 2 \sqrt{ 2 } \right ) \sqrt{ 10 } + \left ( - 2 \sqrt{ 2 } \right ) \times \left ( - \left ( 2 \sqrt{ 2 } \right ) \right ) } { 10 - 8 }$
$ $ Multiply $ 10 $ and $ 10$
$\dfrac { \sqrt{ \color{#FF6800}{ 100 } } + \sqrt{ 10 } \left ( - \left ( 2 \sqrt{ 2 } \right ) \right ) + \left ( - 2 \sqrt{ 2 } \right ) \sqrt{ 10 } + \left ( - 2 \sqrt{ 2 } \right ) \times \left ( - \left ( 2 \sqrt{ 2 } \right ) \right ) } { 10 - 8 }$
$\dfrac { \sqrt{ 100 } + \sqrt{ 10 } \left ( \color{#FF6800}{ - } \left ( 2 \sqrt{ 2 } \right ) \right ) + \left ( - 2 \sqrt{ 2 } \right ) \sqrt{ 10 } + \left ( - 2 \sqrt{ 2 } \right ) \times \left ( - \left ( 2 \sqrt{ 2 } \right ) \right ) } { 10 - 8 }$
$ $ Move the (-) sign forward $ $
$\dfrac { \sqrt{ 100 } \color{#FF6800}{ - } \sqrt{ 10 } \left ( 2 \sqrt{ 2 } \right ) + \left ( - 2 \sqrt{ 2 } \right ) \sqrt{ 10 } + \left ( - 2 \sqrt{ 2 } \right ) \times \left ( - \left ( 2 \sqrt{ 2 } \right ) \right ) } { 10 - 8 }$
$\dfrac { \sqrt{ 100 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 10 } } \left ( \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \right ) + \left ( - 2 \sqrt{ 2 } \right ) \sqrt{ 10 } + \left ( - 2 \sqrt{ 2 } \right ) \times \left ( - \left ( 2 \sqrt{ 2 } \right ) \right ) } { 10 - 8 }$
$ $ Get rid of unnecessary parentheses $ $
$\dfrac { \sqrt{ 100 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 10 } } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } + \left ( - 2 \sqrt{ 2 } \right ) \sqrt{ 10 } + \left ( - 2 \sqrt{ 2 } \right ) \times \left ( - \left ( 2 \sqrt{ 2 } \right ) \right ) } { 10 - 8 }$
$\dfrac { \sqrt{ 100 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 10 } } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } + \left ( - 2 \sqrt{ 2 } \right ) \sqrt{ 10 } + \left ( - 2 \sqrt{ 2 } \right ) \times \left ( - \left ( 2 \sqrt{ 2 } \right ) \right ) } { 10 - 8 }$
$ $ Simplify the expression $ $
$\dfrac { \sqrt{ 100 } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 5 } } + \left ( - 2 \sqrt{ 2 } \right ) \sqrt{ 10 } + \left ( - 2 \sqrt{ 2 } \right ) \times \left ( - \left ( 2 \sqrt{ 2 } \right ) \right ) } { 10 - 8 }$
$\dfrac { \sqrt{ 100 } - 4 \sqrt{ 5 } + \left ( \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \right ) \sqrt{ \color{#FF6800}{ 10 } } + \left ( - 2 \sqrt{ 2 } \right ) \times \left ( - \left ( 2 \sqrt{ 2 } \right ) \right ) } { 10 - 8 }$
$ $ Get rid of unnecessary parentheses $ $
$\dfrac { \sqrt{ 100 } - 4 \sqrt{ 5 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \sqrt{ \color{#FF6800}{ 10 } } + \left ( - 2 \sqrt{ 2 } \right ) \times \left ( - \left ( 2 \sqrt{ 2 } \right ) \right ) } { 10 - 8 }$
$\dfrac { \sqrt{ 100 } - 4 \sqrt{ 5 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \sqrt{ \color{#FF6800}{ 10 } } + \left ( - 2 \sqrt{ 2 } \right ) \times \left ( - \left ( 2 \sqrt{ 2 } \right ) \right ) } { 10 - 8 }$
$ $ Simplify the expression $ $
$\dfrac { \sqrt{ 100 } - 4 \sqrt{ 5 } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 5 } } + \left ( - 2 \sqrt{ 2 } \right ) \times \left ( - \left ( 2 \sqrt{ 2 } \right ) \right ) } { 10 - 8 }$
$\dfrac { \sqrt{ 100 } - 4 \sqrt{ 5 } - 4 \sqrt{ 5 } + \left ( \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \right ) \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \left ( \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \right ) \right ) } { 10 - 8 }$
$ $ Get rid of unnecessary parentheses $ $
$\dfrac { \sqrt{ 100 } - 4 \sqrt{ 5 } - 4 \sqrt{ 5 } + \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } } { 10 - 8 }$
$\dfrac { \sqrt{ 100 } - 4 \sqrt{ 5 } - 4 \sqrt{ 5 } + \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } } { 10 - 8 }$
$ $ Simplify the expression $ $
$\dfrac { \sqrt{ 100 } - 4 \sqrt{ 5 } - 4 \sqrt{ 5 } + \color{#FF6800}{ 8 } } { 10 - 8 }$
$\dfrac { \sqrt{ 100 } - 4 \sqrt{ 5 } - 4 \sqrt{ 5 } + 8 } { \color{#FF6800}{ 10 } \color{#FF6800}{ - } \color{#FF6800}{ 8 } }$
$ $ Subtract $ 8 $ from $ 10$
$\dfrac { \sqrt{ 100 } - 4 \sqrt{ 5 } - 4 \sqrt{ 5 } + 8 } { \color{#FF6800}{ 2 } }$
$\dfrac { \sqrt{ \color{#FF6800}{ 100 } } - 4 \sqrt{ 5 } - 4 \sqrt{ 5 } + 8 } { 2 }$
$ $ Organize the part that can be taken out of the radical sign inside the square root symbol $ $
$\dfrac { \color{#FF6800}{ 10 } - 4 \sqrt{ 5 } - 4 \sqrt{ 5 } + 8 } { 2 }$
$\dfrac { \color{#FF6800}{ 10 } - 4 \sqrt{ 5 } - 4 \sqrt{ 5 } \color{#FF6800}{ + } \color{#FF6800}{ 8 } } { 2 }$
$ $ Add $ 10 $ and $ 8$
$\dfrac { \color{#FF6800}{ 18 } - 4 \sqrt{ 5 } - 4 \sqrt{ 5 } } { 2 }$
$\dfrac { 18 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 5 } } } { 2 }$
$ $ Calculate between similar terms $ $
$\dfrac { 18 \color{#FF6800}{ - } \color{#FF6800}{ 8 } \sqrt{ \color{#FF6800}{ 5 } } } { 2 }$
$\color{#FF6800}{ \dfrac { \color{#FF6800}{ 18 } \color{#FF6800}{ - } \color{#FF6800}{ 8 } \sqrt{ \color{#FF6800}{ 5 } } } { \color{#FF6800}{ 2 } } }$
$ $ Reduce the fraction $ $
$\color{#FF6800}{ 9 } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 5 } }$
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