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$\dfrac { 7 } { 3 }$
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$\dfrac { \sqrt{ 5 } } { \sqrt{ 5 } + \sqrt{ 2 } } + \dfrac { \sqrt{ 2 } } { \sqrt{ 5 } - \sqrt{ 2 } }$
$ $ Find the conjugate irrational number of denominator $ $
$\color{#FF6800}{ \dfrac { \sqrt{ \color{#FF6800}{ 5 } } } { \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 2 } } } } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } } { \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } } } + \dfrac { \sqrt{ 2 } } { \sqrt{ 5 } - \sqrt{ 2 } }$
$\dfrac { \sqrt{ 5 } } { \sqrt{ 5 } + \sqrt{ 2 } } \times \dfrac { \sqrt{ 5 } - \sqrt{ 2 } } { \sqrt{ 5 } - \sqrt{ 2 } } + \dfrac { \sqrt{ 2 } } { \sqrt{ 5 } - \sqrt{ 2 } }$
$ $ The denominator is multiplied by denominator, and the numerator is multiplied by numerator $ $
$\color{#FF6800}{ \dfrac { \sqrt{ \color{#FF6800}{ 5 } } \left ( \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } \right ) } { \left ( \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 2 } } \right ) \left ( \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } \right ) } } + \dfrac { \sqrt{ 2 } } { \sqrt{ 5 } - \sqrt{ 2 } }$
$\dfrac { \sqrt{ \color{#FF6800}{ 5 } } \left ( \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } \right ) } { \left ( \sqrt{ 5 } + \sqrt{ 2 } \right ) \left ( \sqrt{ 5 } - \sqrt{ 2 } \right ) } + \dfrac { \sqrt{ 2 } } { \sqrt{ 5 } - \sqrt{ 2 } }$
$ $ Multiply each term in parentheses by $ \sqrt{ 5 }$
$\dfrac { \sqrt{ \color{#FF6800}{ 5 } } \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 5 } } \left ( \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } \right ) } { \left ( \sqrt{ 5 } + \sqrt{ 2 } \right ) \left ( \sqrt{ 5 } - \sqrt{ 2 } \right ) } + \dfrac { \sqrt{ 2 } } { \sqrt{ 5 } - \sqrt{ 2 } }$
$\dfrac { \sqrt{ 5 } \sqrt{ 5 } + \sqrt{ 5 } \left ( - \sqrt{ 2 } \right ) } { \left ( \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 2 } } \right ) \left ( \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } \right ) } + \dfrac { \sqrt{ 2 } } { \sqrt{ 5 } - \sqrt{ 2 } }$
$ $ Expand the expression using $ \left(a - b\right)\left(a + b\right) = a^{2} - b^{2}$
$\dfrac { \sqrt{ 5 } \sqrt{ 5 } + \sqrt{ 5 } \left ( - \sqrt{ 2 } \right ) } { \left ( \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \left ( \sqrt{ \color{#FF6800}{ 2 } } \right ) ^ { \color{#FF6800}{ 2 } } } + \dfrac { \sqrt{ 2 } } { \sqrt{ 5 } - \sqrt{ 2 } }$
$\dfrac { \sqrt{ \color{#FF6800}{ 5 } } \sqrt{ \color{#FF6800}{ 5 } } + \sqrt{ 5 } \left ( - \sqrt{ 2 } \right ) } { \left ( \sqrt{ 5 } \right ) ^ { 2 } - \left ( \sqrt{ 2 } \right ) ^ { 2 } } + \dfrac { \sqrt{ 2 } } { \sqrt{ 5 } - \sqrt{ 2 } }$
$ $ Arrange the expression $ $
$\dfrac { \sqrt{ \color{#FF6800}{ 5 } \color{#FF6800}{ \times } \color{#FF6800}{ 5 } } + \sqrt{ 5 } \left ( - \sqrt{ 2 } \right ) } { \left ( \sqrt{ 5 } \right ) ^ { 2 } - \left ( \sqrt{ 2 } \right ) ^ { 2 } } + \dfrac { \sqrt{ 2 } } { \sqrt{ 5 } - \sqrt{ 2 } }$
$\dfrac { \sqrt{ 5 \times 5 } + \sqrt{ 5 } \left ( - \sqrt{ 2 } \right ) } { \left ( \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 2 } } - \left ( \sqrt{ 2 } \right ) ^ { 2 } } + \dfrac { \sqrt{ 2 } } { \sqrt{ 5 } - \sqrt{ 2 } }$
$ $ Calculate power $ $
$\dfrac { \sqrt{ 5 \times 5 } + \sqrt{ 5 } \left ( - \sqrt{ 2 } \right ) } { \color{#FF6800}{ 5 } - \left ( \sqrt{ 2 } \right ) ^ { 2 } } + \dfrac { \sqrt{ 2 } } { \sqrt{ 5 } - \sqrt{ 2 } }$
$\dfrac { \sqrt{ 5 \times 5 } + \sqrt{ 5 } \left ( - \sqrt{ 2 } \right ) } { 5 - \left ( \sqrt{ \color{#FF6800}{ 2 } } \right ) ^ { \color{#FF6800}{ 2 } } } + \dfrac { \sqrt{ 2 } } { \sqrt{ 5 } - \sqrt{ 2 } }$
$ $ Calculate power $ $
$\dfrac { \sqrt{ 5 \times 5 } + \sqrt{ 5 } \left ( - \sqrt{ 2 } \right ) } { 5 - \color{#FF6800}{ 2 } } + \dfrac { \sqrt{ 2 } } { \sqrt{ 5 } - \sqrt{ 2 } }$
$\dfrac { \sqrt{ \color{#FF6800}{ 5 } \color{#FF6800}{ \times } \color{#FF6800}{ 5 } } + \sqrt{ 5 } \left ( - \sqrt{ 2 } \right ) } { 5 - 2 } + \dfrac { \sqrt{ 2 } } { \sqrt{ 5 } - \sqrt{ 2 } }$
$ $ Multiply $ 5 $ and $ 5$
$\dfrac { \sqrt{ \color{#FF6800}{ 25 } } + \sqrt{ 5 } \left ( - \sqrt{ 2 } \right ) } { 5 - 2 } + \dfrac { \sqrt{ 2 } } { \sqrt{ 5 } - \sqrt{ 2 } }$
$\dfrac { \sqrt{ 25 } + \sqrt{ 5 } \left ( \color{#FF6800}{ - } \sqrt{ 2 } \right ) } { 5 - 2 } + \dfrac { \sqrt{ 2 } } { \sqrt{ 5 } - \sqrt{ 2 } }$
$ $ Move the (-) sign forward $ $
$\dfrac { \sqrt{ 25 } \color{#FF6800}{ - } \sqrt{ 5 } \sqrt{ 2 } } { 5 - 2 } + \dfrac { \sqrt{ 2 } } { \sqrt{ 5 } - \sqrt{ 2 } }$
$\dfrac { \sqrt{ 25 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } \sqrt{ \color{#FF6800}{ 2 } } } { 5 - 2 } + \dfrac { \sqrt{ 2 } } { \sqrt{ 5 } - \sqrt{ 2 } }$
$ $ Calculate multiplication $ $
$\dfrac { \sqrt{ 25 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 10 } } } { 5 - 2 } + \dfrac { \sqrt{ 2 } } { \sqrt{ 5 } - \sqrt{ 2 } }$
$\dfrac { \sqrt{ 25 } - \sqrt{ 10 } } { \color{#FF6800}{ 5 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } } + \dfrac { \sqrt{ 2 } } { \sqrt{ 5 } - \sqrt{ 2 } }$
$ $ Subtract $ 2 $ from $ 5$
$\dfrac { \sqrt{ 25 } - \sqrt{ 10 } } { \color{#FF6800}{ 3 } } + \dfrac { \sqrt{ 2 } } { \sqrt{ 5 } - \sqrt{ 2 } }$
$\dfrac { \sqrt{ \color{#FF6800}{ 25 } } - \sqrt{ 10 } } { 3 } + \dfrac { \sqrt{ 2 } } { \sqrt{ 5 } - \sqrt{ 2 } }$
$ $ Organize the part that can be taken out of the radical sign inside the square root symbol $ $
$\dfrac { \color{#FF6800}{ 5 } - \sqrt{ 10 } } { 3 } + \dfrac { \sqrt{ 2 } } { \sqrt{ 5 } - \sqrt{ 2 } }$
$\dfrac { 5 - \sqrt{ 10 } } { 3 } + \dfrac { \sqrt{ 2 } } { \sqrt{ 5 } - \sqrt{ 2 } }$
$ $ Find the conjugate irrational number of denominator $ $
$\dfrac { 5 - \sqrt{ 10 } } { 3 } + \color{#FF6800}{ \dfrac { \sqrt{ \color{#FF6800}{ 2 } } } { \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } } } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 2 } } } { \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 2 } } } }$
$\dfrac { 5 - \sqrt{ 10 } } { 3 } + \dfrac { \sqrt{ 2 } } { \sqrt{ 5 } - \sqrt{ 2 } } \times \dfrac { \sqrt{ 5 } + \sqrt{ 2 } } { \sqrt{ 5 } + \sqrt{ 2 } }$
$ $ The denominator is multiplied by denominator, and the numerator is multiplied by numerator $ $
$\dfrac { 5 - \sqrt{ 10 } } { 3 } + \color{#FF6800}{ \dfrac { \sqrt{ \color{#FF6800}{ 2 } } \left ( \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 2 } } \right ) } { \left ( \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } \right ) \left ( \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 2 } } \right ) } }$
$\dfrac { 5 - \sqrt{ 10 } } { 3 } + \dfrac { \sqrt{ \color{#FF6800}{ 2 } } \left ( \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 2 } } \right ) } { \left ( \sqrt{ 5 } - \sqrt{ 2 } \right ) \left ( \sqrt{ 5 } + \sqrt{ 2 } \right ) }$
$ $ Multiply each term in parentheses by $ \sqrt{ 2 }$
$\dfrac { 5 - \sqrt{ 10 } } { 3 } + \dfrac { \sqrt{ \color{#FF6800}{ 2 } } \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 2 } } \sqrt{ \color{#FF6800}{ 2 } } } { \left ( \sqrt{ 5 } - \sqrt{ 2 } \right ) \left ( \sqrt{ 5 } + \sqrt{ 2 } \right ) }$
$\dfrac { 5 - \sqrt{ 10 } } { 3 } + \dfrac { \sqrt{ 2 } \sqrt{ 5 } + \sqrt{ 2 } \sqrt{ 2 } } { \left ( \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } \right ) \left ( \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 2 } } \right ) }$
$ $ Expand the expression using $ \left(a - b\right)\left(a + b\right) = a^{2} - b^{2}$
$\dfrac { 5 - \sqrt{ 10 } } { 3 } + \dfrac { \sqrt{ 2 } \sqrt{ 5 } + \sqrt{ 2 } \sqrt{ 2 } } { \left ( \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \left ( \sqrt{ \color{#FF6800}{ 2 } } \right ) ^ { \color{#FF6800}{ 2 } } }$
$\dfrac { 5 - \sqrt{ 10 } } { 3 } + \dfrac { \sqrt{ \color{#FF6800}{ 2 } } \sqrt{ \color{#FF6800}{ 5 } } + \sqrt{ 2 } \sqrt{ 2 } } { \left ( \sqrt{ 5 } \right ) ^ { 2 } - \left ( \sqrt{ 2 } \right ) ^ { 2 } }$
$ $ Arrange the expression $ $
$\dfrac { 5 - \sqrt{ 10 } } { 3 } + \dfrac { \sqrt{ \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 5 } } + \sqrt{ 2 } \sqrt{ 2 } } { \left ( \sqrt{ 5 } \right ) ^ { 2 } - \left ( \sqrt{ 2 } \right ) ^ { 2 } }$
$\dfrac { 5 - \sqrt{ 10 } } { 3 } + \dfrac { \sqrt{ 2 \times 5 } + \sqrt{ \color{#FF6800}{ 2 } } \sqrt{ \color{#FF6800}{ 2 } } } { \left ( \sqrt{ 5 } \right ) ^ { 2 } - \left ( \sqrt{ 2 } \right ) ^ { 2 } }$
$ $ Arrange the expression $ $
$\dfrac { 5 - \sqrt{ 10 } } { 3 } + \dfrac { \sqrt{ 2 \times 5 } + \sqrt{ \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } } } { \left ( \sqrt{ 5 } \right ) ^ { 2 } - \left ( \sqrt{ 2 } \right ) ^ { 2 } }$
$\dfrac { 5 - \sqrt{ 10 } } { 3 } + \dfrac { \sqrt{ 2 \times 5 } + \sqrt{ 2 \times 2 } } { \left ( \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 2 } } - \left ( \sqrt{ 2 } \right ) ^ { 2 } }$
$ $ Calculate power $ $
$\dfrac { 5 - \sqrt{ 10 } } { 3 } + \dfrac { \sqrt{ 2 \times 5 } + \sqrt{ 2 \times 2 } } { \color{#FF6800}{ 5 } - \left ( \sqrt{ 2 } \right ) ^ { 2 } }$
$\dfrac { 5 - \sqrt{ 10 } } { 3 } + \dfrac { \sqrt{ 2 \times 5 } + \sqrt{ 2 \times 2 } } { 5 - \left ( \sqrt{ \color{#FF6800}{ 2 } } \right ) ^ { \color{#FF6800}{ 2 } } }$
$ $ Calculate power $ $
$\dfrac { 5 - \sqrt{ 10 } } { 3 } + \dfrac { \sqrt{ 2 \times 5 } + \sqrt{ 2 \times 2 } } { 5 - \color{#FF6800}{ 2 } }$
$\dfrac { 5 - \sqrt{ 10 } } { 3 } + \dfrac { \sqrt{ \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 5 } } + \sqrt{ 2 \times 2 } } { 5 - 2 }$
$ $ Multiply $ 2 $ and $ 5$
$\dfrac { 5 - \sqrt{ 10 } } { 3 } + \dfrac { \sqrt{ \color{#FF6800}{ 10 } } + \sqrt{ 2 \times 2 } } { 5 - 2 }$
$\dfrac { 5 - \sqrt{ 10 } } { 3 } + \dfrac { \sqrt{ 10 } + \sqrt{ \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } } } { 5 - 2 }$
$ $ Multiply $ 2 $ and $ 2$
$\dfrac { 5 - \sqrt{ 10 } } { 3 } + \dfrac { \sqrt{ 10 } + \sqrt{ \color{#FF6800}{ 4 } } } { 5 - 2 }$
$\dfrac { 5 - \sqrt{ 10 } } { 3 } + \dfrac { \sqrt{ 10 } + \sqrt{ 4 } } { \color{#FF6800}{ 5 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } }$
$ $ Subtract $ 2 $ from $ 5$
$\dfrac { 5 - \sqrt{ 10 } } { 3 } + \dfrac { \sqrt{ 10 } + \sqrt{ 4 } } { \color{#FF6800}{ 3 } }$
$\dfrac { 5 - \sqrt{ 10 } } { 3 } + \dfrac { \sqrt{ 10 } + \sqrt{ \color{#FF6800}{ 4 } } } { 3 }$
$ $ Organize the part that can be taken out of the radical sign inside the square root symbol $ $
$\dfrac { 5 - \sqrt{ 10 } } { 3 } + \dfrac { \sqrt{ 10 } + \color{#FF6800}{ 2 } } { 3 }$
$\color{#FF6800}{ \dfrac { \color{#FF6800}{ 5 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 10 } } } { \color{#FF6800}{ 3 } } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { \sqrt{ \color{#FF6800}{ 10 } } \color{#FF6800}{ + } \color{#FF6800}{ 2 } } { \color{#FF6800}{ 3 } } }$
$ $ Combine the fraction with the same denominator $ $
$\dfrac { \color{#FF6800}{ 5 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 10 } } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 10 } } \color{#FF6800}{ + } \color{#FF6800}{ 2 } } { 3 }$
$\dfrac { 5 \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 10 } } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 10 } } + 2 } { 3 }$
$ $ Remove the two numbers if the values are the same and the signs are different $ $
$\dfrac { 5 + 2 } { 3 }$
$\dfrac { \color{#FF6800}{ 5 } \color{#FF6800}{ + } \color{#FF6800}{ 2 } } { 3 }$
$ $ Add $ 5 $ and $ 2$
$\dfrac { \color{#FF6800}{ 7 } } { 3 }$
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