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$3 - 2 \sqrt{ 2 }$
Calculate the value
$\dfrac { \sqrt{ 2 } - 1 } { \sqrt{ 2 } + 1 }$
$ $ Find the conjugate irrational number of denominator $ $
$\color{#FF6800}{ \dfrac { \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 } } { \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 } } } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 } } { \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 } } }$
$\dfrac { \sqrt{ 2 } - 1 } { \sqrt{ 2 } + 1 } \times \dfrac { \sqrt{ 2 } - 1 } { \sqrt{ 2 } - 1 }$
$ $ The denominator is multiplied by denominator, and the numerator is multiplied by numerator $ $
$\color{#FF6800}{ \dfrac { \left ( \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) \left ( \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) } { \left ( \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) \left ( \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) } }$
$\dfrac { \left ( \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) \left ( \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) } { \left ( \sqrt{ 2 } + 1 \right ) \left ( \sqrt{ 2 } - 1 \right ) }$
$ $ Expand using $ \left(a + b\right)\left(c + d\right) = ac + ad + bc + bd$
$\dfrac { \sqrt{ \color{#FF6800}{ 2 } } \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 2 } } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) \color{#FF6800}{ - } \color{#FF6800}{ 1 } \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) } { \left ( \sqrt{ 2 } + 1 \right ) \left ( \sqrt{ 2 } - 1 \right ) }$
$\dfrac { \sqrt{ 2 } \sqrt{ 2 } + \sqrt{ 2 } \left ( - 1 \right ) - 1 \sqrt{ 2 } - 1 \times \left ( - 1 \right ) } { \left ( \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) \left ( \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) }$
$ $ Expand the expression using $ \left(a - b\right)\left(a + b\right) = a^{2} - b^{2}$
$\dfrac { \sqrt{ 2 } \sqrt{ 2 } + \sqrt{ 2 } \left ( - 1 \right ) - 1 \sqrt{ 2 } - 1 \times \left ( - 1 \right ) } { \left ( \sqrt{ \color{#FF6800}{ 2 } } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 } ^ { \color{#FF6800}{ 2 } } }$
$\dfrac { \sqrt{ \color{#FF6800}{ 2 } } \sqrt{ \color{#FF6800}{ 2 } } + \sqrt{ 2 } \left ( - 1 \right ) - 1 \sqrt{ 2 } - 1 \times \left ( - 1 \right ) } { \left ( \sqrt{ 2 } \right ) ^ { 2 } - 1 ^ { 2 } }$
$ $ Arrange the expression $ $
$\dfrac { \sqrt{ \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } } + \sqrt{ 2 } \left ( - 1 \right ) - 1 \sqrt{ 2 } - 1 \times \left ( - 1 \right ) } { \left ( \sqrt{ 2 } \right ) ^ { 2 } - 1 ^ { 2 } }$
$\dfrac { \sqrt{ 2 \times 2 } + \sqrt{ 2 } \left ( - 1 \right ) - 1 \sqrt{ 2 } - 1 \times \left ( - 1 \right ) } { \left ( \sqrt{ \color{#FF6800}{ 2 } } \right ) ^ { \color{#FF6800}{ 2 } } - 1 ^ { 2 } }$
$ $ Calculate power $ $
$\dfrac { \sqrt{ 2 \times 2 } + \sqrt{ 2 } \left ( - 1 \right ) - 1 \sqrt{ 2 } - 1 \times \left ( - 1 \right ) } { \color{#FF6800}{ 2 } - 1 ^ { 2 } }$
$\dfrac { \sqrt{ 2 \times 2 } + \sqrt{ 2 } \left ( - 1 \right ) - 1 \sqrt{ 2 } - 1 \times \left ( - 1 \right ) } { 2 - \color{#FF6800}{ 1 } ^ { \color{#FF6800}{ 2 } } }$
$ $ Calculate power $ $
$\dfrac { \sqrt{ 2 \times 2 } + \sqrt{ 2 } \left ( - 1 \right ) - 1 \sqrt{ 2 } - 1 \times \left ( - 1 \right ) } { 2 - \color{#FF6800}{ 1 } }$
$\dfrac { \sqrt{ \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } } + \sqrt{ 2 } \left ( - 1 \right ) - 1 \sqrt{ 2 } - 1 \times \left ( - 1 \right ) } { 2 - 1 }$
$ $ Multiply $ 2 $ and $ 2$
$\dfrac { \sqrt{ \color{#FF6800}{ 4 } } + \sqrt{ 2 } \left ( - 1 \right ) - 1 \sqrt{ 2 } - 1 \times \left ( - 1 \right ) } { 2 - 1 }$
$\dfrac { \sqrt{ 4 } + \sqrt{ \color{#FF6800}{ 2 } } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) - 1 \sqrt{ 2 } - 1 \times \left ( - 1 \right ) } { 2 - 1 }$
$ $ Simplify the expression $ $
$\dfrac { \sqrt{ 4 } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \sqrt{ \color{#FF6800}{ 2 } } - 1 \sqrt{ 2 } - 1 \times \left ( - 1 \right ) } { 2 - 1 }$
$\dfrac { \sqrt{ 4 } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \sqrt{ 2 } - 1 \sqrt{ 2 } - 1 \times \left ( - 1 \right ) } { 2 - 1 }$
$ $ Multiplying any number by 1 does not change the value $ $
$\dfrac { \sqrt{ 4 } - \sqrt{ 2 } - 1 \sqrt{ 2 } - 1 \times \left ( - 1 \right ) } { 2 - 1 }$
$\dfrac { \sqrt{ 4 } - \sqrt{ 2 } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \sqrt{ 2 } - 1 \times \left ( - 1 \right ) } { 2 - 1 }$
$ $ Multiplying any number by 1 does not change the value $ $
$\dfrac { \sqrt{ 4 } - \sqrt{ 2 } - \sqrt{ 2 } - 1 \times \left ( - 1 \right ) } { 2 - 1 }$
$\dfrac { \sqrt{ 4 } - \sqrt{ 2 } - \sqrt{ 2 } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) } { 2 - 1 }$
$ $ Multiply $ - 1 $ and $ - 1$
$\dfrac { \sqrt{ 4 } - \sqrt{ 2 } - \sqrt{ 2 } + \color{#FF6800}{ 1 } } { 2 - 1 }$
$\dfrac { \sqrt{ 4 } - \sqrt{ 2 } - \sqrt{ 2 } + 1 } { \color{#FF6800}{ 2 } \color{#FF6800}{ - } \color{#FF6800}{ 1 } }$
$ $ Subtract $ 1 $ from $ 2$
$\dfrac { \sqrt{ 4 } - \sqrt{ 2 } - \sqrt{ 2 } + 1 } { \color{#FF6800}{ 1 } }$
$\dfrac { \sqrt{ 4 } - \sqrt{ 2 } - \sqrt{ 2 } + 1 } { \color{#FF6800}{ 1 } }$
$ $ If the denominator is 1, the denominator can be removed $ $
$\sqrt{ \color{#FF6800}{ 4 } } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 }$
$\sqrt{ \color{#FF6800}{ 4 } } - \sqrt{ 2 } - \sqrt{ 2 } + 1$
$ $ Organize the part that can be taken out of the radical sign inside the square root symbol $ $
$\color{#FF6800}{ 2 } - \sqrt{ 2 } - \sqrt{ 2 } + 1$
$\color{#FF6800}{ 2 } - \sqrt{ 2 } - \sqrt{ 2 } \color{#FF6800}{ + } \color{#FF6800}{ 1 }$
$ $ Add $ 2 $ and $ 1$
$\color{#FF6800}{ 3 } - \sqrt{ 2 } - \sqrt{ 2 }$
$3 \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } }$
$ $ Calculate between similar terms $ $
$3 \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } }$
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