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Answer
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$\dfrac{ 2- \sqrt{ 5 } }{ 2+ \sqrt{ 5 } } + \dfrac{ 2+ \sqrt{ 5 } }{ 2- \sqrt{ 5 } }$
$- 18$
Calculate the value
$\dfrac { 2 - \sqrt{ 5 } } { 2 + \sqrt{ 5 } } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$ $ Find the conjugate irrational number of denominator $ $
$\color{#FF6800}{ \dfrac { 2 - \sqrt{ 5 } } { 2 + \sqrt{ 5 } } } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 2 - \sqrt{ 5 } } { 2 - \sqrt{ 5 } } } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$\dfrac { 2 - \sqrt{ 5 } } { 2 + \sqrt{ 5 } } \times \dfrac { 2 - \sqrt{ 5 } } { 2 - \sqrt{ 5 } } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$ $ The denominator is multiplied by denominator, and the numerator is multiplied by numerator $ $
$\color{#FF6800}{ \dfrac { \left ( 2 - \sqrt{ 5 } \right ) \left ( 2 - \sqrt{ 5 } \right ) } { \left ( 2 + \sqrt{ 5 } \right ) \left ( 2 - \sqrt{ 5 } \right ) } } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$\dfrac { \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } \right ) \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } \right ) } { \left ( 2 + \sqrt{ 5 } \right ) \left ( 2 - \sqrt{ 5 } \right ) } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$ $ Expand using $ \left(a + b\right)\left(c + d\right) = ac + ad + bc + bd$
$\dfrac { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } \right ) \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } \right ) } { \left ( 2 + \sqrt{ 5 } \right ) \left ( 2 - \sqrt{ 5 } \right ) } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$\dfrac { 2 \times 2 + 2 \times \left ( - \sqrt{ 5 } \right ) - \sqrt{ 5 } \times 2 - \sqrt{ 5 } \times \left ( - \sqrt{ 5 } \right ) } { \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 5 } } \right ) \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } \right ) } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$ $ Expand the expression using $ \left(a - b\right)\left(a + b\right) = a^{2} - b^{2}$
$\dfrac { 2 \times 2 + 2 \times \left ( - \sqrt{ 5 } \right ) - \sqrt{ 5 } \times 2 - \sqrt{ 5 } \times \left ( - \sqrt{ 5 } \right ) } { \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \left ( \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 2 } } } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$\dfrac { 2 \times 2 + 2 \times \left ( - \sqrt{ 5 } \right ) - \sqrt{ 5 } \times 2 - \sqrt{ 5 } \times \left ( - \sqrt{ 5 } \right ) } { \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } - \left ( \sqrt{ 5 } \right ) ^ { 2 } } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$ $ Calculate power $ $
$\dfrac { 2 \times 2 + 2 \times \left ( - \sqrt{ 5 } \right ) - \sqrt{ 5 } \times 2 - \sqrt{ 5 } \times \left ( - \sqrt{ 5 } \right ) } { \color{#FF6800}{ 4 } - \left ( \sqrt{ 5 } \right ) ^ { 2 } } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$\dfrac { 2 \times 2 + 2 \times \left ( - \sqrt{ 5 } \right ) - \sqrt{ 5 } \times 2 - \sqrt{ 5 } \times \left ( - \sqrt{ 5 } \right ) } { 4 - \left ( \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 2 } } } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$ $ Calculate power $ $
$\dfrac { 2 \times 2 + 2 \times \left ( - \sqrt{ 5 } \right ) - \sqrt{ 5 } \times 2 - \sqrt{ 5 } \times \left ( - \sqrt{ 5 } \right ) } { 4 - \color{#FF6800}{ 5 } } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$\dfrac { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } + 2 \times \left ( - \sqrt{ 5 } \right ) - \sqrt{ 5 } \times 2 - \sqrt{ 5 } \times \left ( - \sqrt{ 5 } \right ) } { 4 - 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$ $ Multiply $ 2 $ and $ 2$
$\dfrac { \color{#FF6800}{ 4 } + 2 \times \left ( - \sqrt{ 5 } \right ) - \sqrt{ 5 } \times 2 - \sqrt{ 5 } \times \left ( - \sqrt{ 5 } \right ) } { 4 - 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$\dfrac { 4 + \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } \right ) - \sqrt{ 5 } \times 2 - \sqrt{ 5 } \times \left ( - \sqrt{ 5 } \right ) } { 4 - 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$ $ Simplify the expression $ $
$\dfrac { 4 \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 5 } } - \sqrt{ 5 } \times 2 - \sqrt{ 5 } \times \left ( - \sqrt{ 5 } \right ) } { 4 - 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$\dfrac { 4 - 2 \sqrt{ 5 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } - \sqrt{ 5 } \times \left ( - \sqrt{ 5 } \right ) } { 4 - 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$ $ Simplify the expression $ $
$\dfrac { 4 - 2 \sqrt{ 5 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 5 } } - \sqrt{ 5 } \times \left ( - \sqrt{ 5 } \right ) } { 4 - 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$\dfrac { 4 - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } \color{#FF6800}{ - } \sqrt{ 5 } \times \left ( \color{#FF6800}{ - } \sqrt{ 5 } \right ) } { 4 - 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$ $ Since negative numbers are multiplied by an even number, remove the (-) sign $ $
$\dfrac { 4 - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } + \sqrt{ 5 } \sqrt{ 5 } } { 4 - 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$\dfrac { 4 - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } + \sqrt{ \color{#FF6800}{ 5 } } \sqrt{ 5 } } { 4 - 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$ $ If the exponent is omitted, the exponent of that term is equal to 1 $ $
$\dfrac { 4 - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } + \left ( \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 1 } } \sqrt{ 5 } } { 4 - 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$\dfrac { 4 - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } + \left ( \sqrt{ 5 } \right ) ^ { 1 } \sqrt{ \color{#FF6800}{ 5 } } } { 4 - 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$ $ If the exponent is omitted, the exponent of that term is equal to 1 $ $
$\dfrac { 4 - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } + \left ( \sqrt{ 5 } \right ) ^ { 1 } \left ( \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 1 } } } { 4 - 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$\dfrac { 4 - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } + \left ( \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 1 } } \left ( \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 1 } } } { 4 - 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$ $ Add the exponent as the base is the same $ $
$\dfrac { 4 - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } + \left ( \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } } } { 4 - 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$\dfrac { 4 - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } + \left ( \sqrt{ 5 } \right ) ^ { \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } } } { 4 - 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$ $ Add $ 1 $ and $ 1$
$\dfrac { 4 - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } + \left ( \sqrt{ 5 } \right ) ^ { \color{#FF6800}{ 2 } } } { 4 - 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$\dfrac { 4 - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } + \left ( \sqrt{ 5 } \right ) ^ { 2 } } { \color{#FF6800}{ 4 } \color{#FF6800}{ - } \color{#FF6800}{ 5 } } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$ $ Subtract $ 5 $ from $ 4$
$\dfrac { 4 - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } + \left ( \sqrt{ 5 } \right ) ^ { 2 } } { \color{#FF6800}{ - } \color{#FF6800}{ 1 } } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$\dfrac { 4 - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } + \left ( \sqrt{ 5 } \right ) ^ { 2 } } { \color{#FF6800}{ - } 1 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$ $ If the denominator is 1, the denominator can be removed $ $
$\color{#FF6800}{ - } \left ( \color{#FF6800}{ 4 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ + } \left ( \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 2 } } \right ) + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$- \left ( 4 - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } + \left ( \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 2 } } \right ) + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$ $ If you square the radical sign, it will disappear $ $
$- \left ( 4 - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } + \color{#FF6800}{ 5 } \right ) + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$- \left ( \color{#FF6800}{ 4 } - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } \color{#FF6800}{ + } \color{#FF6800}{ 5 } \right ) + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$ $ Add $ 4 $ and $ 5$
$- \left ( \color{#FF6800}{ 9 } - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } \right ) + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$- \left ( 9 \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 5 } } \right ) + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$ $ Calculate between similar terms $ $
$- \left ( 9 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 5 } } \right ) + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$\color{#FF6800}{ - } \left ( \color{#FF6800}{ 9 } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 5 } } \right ) + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$ $ Change the symbol of each term in parentheses when there is a (-) symbol in front of parentheses $ $
$\color{#FF6800}{ - } \color{#FF6800}{ 9 } + \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 5 } } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$- 9 + 4 \sqrt{ 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$ $ Find the conjugate irrational number of denominator $ $
$- 9 + 4 \sqrt{ 5 } + \color{#FF6800}{ \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } } } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 2 + \sqrt{ 5 } } { 2 + \sqrt{ 5 } } }$
$- 9 + 4 \sqrt{ 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } } \times \dfrac { 2 + \sqrt{ 5 } } { 2 + \sqrt{ 5 } }$
$ $ The denominator is multiplied by denominator, and the numerator is multiplied by numerator $ $
$- 9 + 4 \sqrt{ 5 } + \color{#FF6800}{ \dfrac { \left ( 2 + \sqrt{ 5 } \right ) \left ( 2 + \sqrt{ 5 } \right ) } { \left ( 2 - \sqrt{ 5 } \right ) \left ( 2 + \sqrt{ 5 } \right ) } }$
$- 9 + 4 \sqrt{ 5 } + \dfrac { \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 5 } } \right ) \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 5 } } \right ) } { \left ( 2 - \sqrt{ 5 } \right ) \left ( 2 + \sqrt{ 5 } \right ) }$
$ $ Expand using $ \left(a + b\right)\left(c + d\right) = ac + ad + bc + bd$
$- 9 + 4 \sqrt{ 5 } + \dfrac { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 5 } } \sqrt{ \color{#FF6800}{ 5 } } } { \left ( 2 - \sqrt{ 5 } \right ) \left ( 2 + \sqrt{ 5 } \right ) }$
$- 9 + 4 \sqrt{ 5 } + \dfrac { 2 \times 2 + 2 \sqrt{ 5 } + \sqrt{ 5 } \times 2 + \sqrt{ 5 } \sqrt{ 5 } } { \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } \right ) \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 5 } } \right ) }$
$ $ Expand the expression using $ \left(a - b\right)\left(a + b\right) = a^{2} - b^{2}$
$- 9 + 4 \sqrt{ 5 } + \dfrac { 2 \times 2 + 2 \sqrt{ 5 } + \sqrt{ 5 } \times 2 + \sqrt{ 5 } \sqrt{ 5 } } { \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \left ( \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 2 } } }$
$- 9 + 4 \sqrt{ 5 } + \dfrac { 2 \times 2 + 2 \sqrt{ 5 } + \sqrt{ 5 } \times 2 + \sqrt{ \color{#FF6800}{ 5 } } \sqrt{ \color{#FF6800}{ 5 } } } { 2 ^ { 2 } - \left ( \sqrt{ 5 } \right ) ^ { 2 } }$
$ $ Arrange the expression $ $
$- 9 + 4 \sqrt{ 5 } + \dfrac { 2 \times 2 + 2 \sqrt{ 5 } + \sqrt{ 5 } \times 2 + \sqrt{ \color{#FF6800}{ 5 } \color{#FF6800}{ \times } \color{#FF6800}{ 5 } } } { 2 ^ { 2 } - \left ( \sqrt{ 5 } \right ) ^ { 2 } }$
$- 9 + 4 \sqrt{ 5 } + \dfrac { 2 \times 2 + 2 \sqrt{ 5 } + \sqrt{ 5 } \times 2 + \sqrt{ 5 \times 5 } } { \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } - \left ( \sqrt{ 5 } \right ) ^ { 2 } }$
$ $ Calculate power $ $
$- 9 + 4 \sqrt{ 5 } + \dfrac { 2 \times 2 + 2 \sqrt{ 5 } + \sqrt{ 5 } \times 2 + \sqrt{ 5 \times 5 } } { \color{#FF6800}{ 4 } - \left ( \sqrt{ 5 } \right ) ^ { 2 } }$
$- 9 + 4 \sqrt{ 5 } + \dfrac { 2 \times 2 + 2 \sqrt{ 5 } + \sqrt{ 5 } \times 2 + \sqrt{ 5 \times 5 } } { 4 - \left ( \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 2 } } }$
$ $ Calculate power $ $
$- 9 + 4 \sqrt{ 5 } + \dfrac { 2 \times 2 + 2 \sqrt{ 5 } + \sqrt{ 5 } \times 2 + \sqrt{ 5 \times 5 } } { 4 - \color{#FF6800}{ 5 } }$
$- 9 + 4 \sqrt{ 5 } + \dfrac { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } + 2 \sqrt{ 5 } + \sqrt{ 5 } \times 2 + \sqrt{ 5 \times 5 } } { 4 - 5 }$
$ $ Multiply $ 2 $ and $ 2$
$- 9 + 4 \sqrt{ 5 } + \dfrac { \color{#FF6800}{ 4 } + 2 \sqrt{ 5 } + \sqrt{ 5 } \times 2 + \sqrt{ 5 \times 5 } } { 4 - 5 }$
$- 9 + 4 \sqrt{ 5 } + \dfrac { 4 + 2 \sqrt{ 5 } + \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } + \sqrt{ 5 \times 5 } } { 4 - 5 }$
$ $ Simplify the expression $ $
$- 9 + 4 \sqrt{ 5 } + \dfrac { 4 + 2 \sqrt{ 5 } + \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 5 } } + \sqrt{ 5 \times 5 } } { 4 - 5 }$
$- 9 + 4 \sqrt{ 5 } + \dfrac { 4 + 2 \sqrt{ 5 } + 2 \sqrt{ 5 } + \sqrt{ \color{#FF6800}{ 5 } \color{#FF6800}{ \times } \color{#FF6800}{ 5 } } } { 4 - 5 }$
$ $ Multiply $ 5 $ and $ 5$
$- 9 + 4 \sqrt{ 5 } + \dfrac { 4 + 2 \sqrt{ 5 } + 2 \sqrt{ 5 } + \sqrt{ \color{#FF6800}{ 25 } } } { 4 - 5 }$
$- 9 + 4 \sqrt{ 5 } + \dfrac { 4 + 2 \sqrt{ 5 } + 2 \sqrt{ 5 } + \sqrt{ 25 } } { \color{#FF6800}{ 4 } \color{#FF6800}{ - } \color{#FF6800}{ 5 } }$
$ $ Subtract $ 5 $ from $ 4$
$- 9 + 4 \sqrt{ 5 } + \dfrac { 4 + 2 \sqrt{ 5 } + 2 \sqrt{ 5 } + \sqrt{ 25 } } { \color{#FF6800}{ - } \color{#FF6800}{ 1 } }$
$- 9 + 4 \sqrt{ 5 } + \dfrac { 4 + 2 \sqrt{ 5 } + 2 \sqrt{ 5 } + \sqrt{ 25 } } { \color{#FF6800}{ - } 1 }$
$ $ If the denominator is 1, the denominator can be removed $ $
$- 9 + 4 \sqrt{ 5 } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 4 } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 25 } } \right )$
$- 9 + 4 \sqrt{ 5 } - \left ( 4 + 2 \sqrt{ 5 } + 2 \sqrt{ 5 } + \sqrt{ \color{#FF6800}{ 25 } } \right )$
$ $ Organize the part that can be taken out of the radical sign inside the square root symbol $ $
$- 9 + 4 \sqrt{ 5 } - \left ( 4 + 2 \sqrt{ 5 } + 2 \sqrt{ 5 } + \color{#FF6800}{ 5 } \right )$
$- 9 + 4 \sqrt{ 5 } - \left ( \color{#FF6800}{ 4 } + 2 \sqrt{ 5 } + 2 \sqrt{ 5 } \color{#FF6800}{ + } \color{#FF6800}{ 5 } \right )$
$ $ Add $ 4 $ and $ 5$
$- 9 + 4 \sqrt{ 5 } - \left ( \color{#FF6800}{ 9 } + 2 \sqrt{ 5 } + 2 \sqrt{ 5 } \right )$
$- 9 + 4 \sqrt{ 5 } - \left ( 9 + \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 5 } } \right )$
$ $ Calculate between similar terms $ $
$- 9 + 4 \sqrt{ 5 } - \left ( 9 + \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 5 } } \right )$
$- 9 + 4 \sqrt{ 5 } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 9 } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 5 } } \right )$
$ $ Change the symbol of each term in parentheses when there is a (-) symbol in front of parentheses $ $
$- 9 + 4 \sqrt{ 5 } \color{#FF6800}{ - } \color{#FF6800}{ 9 } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 5 } }$
$- 9 \color{#FF6800}{ + } \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 5 } } - 9 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 5 } }$
$ $ Eliminate opponent number $ $
$- 9 - 9$
$\color{#FF6800}{ - } \color{#FF6800}{ 9 } \color{#FF6800}{ - } \color{#FF6800}{ 9 }$
$ $ Find the sum of the negative numbers $ $
$\color{#FF6800}{ - } \color{#FF6800}{ 18 }$
Solution search results
search-thumbnail-Simplify $\dfrac {7+\sqrt{5} } {2+\sqrt{5} }+\dfrac {7-\sqrt{5} } {2-\sqrt{5} }$
7th-9th grade
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search-thumbnail-Simplify $\dfrac {2+\sqrt{5} } {2-\sqrt{5} }+\dfrac {2-\sqrt{5} } {2+\sqrt{5} }$
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search-thumbnail-$2$ 
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search-thumbnail-$\dfrac {2+\sqrt{5} } {2-\sqrt{5} }$ $+\dfrac {2-\sqrt{5} } {2+\sqrt{5} }$ $+3-8$ $3$ $1$
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search-thumbnail-The rationalizing factor of \sqrt{23} is 
$°$ $Options^{°}$ $0$ 
A 24 
23 
C \sqrt{23} 
D None of these
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