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Formula
Calculate the value
$\dfrac{ 1 }{ \sqrt{ 2 } -1 } + \dfrac{ 1 }{ \sqrt{ 2 } +1 }$
$2 \sqrt{ 2 }$
Calculate the value
$\dfrac { 1 } { \sqrt{ 2 } - 1 } + \dfrac { 1 } { \sqrt{ 2 } + 1 }$
 Find the conjugate irrational number of denominator 
$\color{#FF6800}{ \dfrac { 1 } { \sqrt{ 2 } - 1 } } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { \sqrt{ 2 } + 1 } { \sqrt{ 2 } + 1 } } + \dfrac { 1 } { \sqrt{ 2 } + 1 }$
$\dfrac { 1 } { \sqrt{ 2 } - 1 } \times \dfrac { \sqrt{ 2 } + 1 } { \sqrt{ 2 } + 1 } + \dfrac { 1 } { \sqrt{ 2 } + 1 }$
 The denominator is multiplied by denominator, and the numerator is multiplied by numerator 
$\color{#FF6800}{ \dfrac { 1 \left ( \sqrt{ 2 } + 1 \right ) } { \left ( \sqrt{ 2 } - 1 \right ) \left ( \sqrt{ 2 } + 1 \right ) } } + \dfrac { 1 } { \sqrt{ 2 } + 1 }$
$\dfrac { \color{#FF6800}{ 1 } \left ( \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) } { \left ( \sqrt{ 2 } - 1 \right ) \left ( \sqrt{ 2 } + 1 \right ) } + \dfrac { 1 } { \sqrt{ 2 } + 1 }$
 Multiply each term in parentheses by $1$
$\dfrac { \color{#FF6800}{ 1 } \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 } } { \left ( \sqrt{ 2 } - 1 \right ) \left ( \sqrt{ 2 } + 1 \right ) } + \dfrac { 1 } { \sqrt{ 2 } + 1 }$
$\dfrac { 1 \sqrt{ 2 } + 1 } { \left ( \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) \left ( \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) } + \dfrac { 1 } { \sqrt{ 2 } + 1 }$
 Expand the expression using $\left(a - b\right)\left(a + b\right) = a^{2} - b^{2}$
$\dfrac { 1 \sqrt{ 2 } + 1 } { \left ( \sqrt{ \color{#FF6800}{ 2 } } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 } ^ { \color{#FF6800}{ 2 } } } + \dfrac { 1 } { \sqrt{ 2 } + 1 }$
$\dfrac { 1 \sqrt{ 2 } + 1 } { \left ( \sqrt{ \color{#FF6800}{ 2 } } \right ) ^ { \color{#FF6800}{ 2 } } - 1 ^ { 2 } } + \dfrac { 1 } { \sqrt{ 2 } + 1 }$
 Calculate power 
$\dfrac { 1 \sqrt{ 2 } + 1 } { \color{#FF6800}{ 2 } - 1 ^ { 2 } } + \dfrac { 1 } { \sqrt{ 2 } + 1 }$
$\dfrac { 1 \sqrt{ 2 } + 1 } { 2 - \color{#FF6800}{ 1 } ^ { \color{#FF6800}{ 2 } } } + \dfrac { 1 } { \sqrt{ 2 } + 1 }$
 Calculate power 
$\dfrac { 1 \sqrt{ 2 } + 1 } { 2 - \color{#FF6800}{ 1 } } + \dfrac { 1 } { \sqrt{ 2 } + 1 }$
$\dfrac { \color{#FF6800}{ 1 } \sqrt{ 2 } + 1 } { 2 - 1 } + \dfrac { 1 } { \sqrt{ 2 } + 1 }$
 Multiplying any number by 1 does not change the value 
$\dfrac { \sqrt{ 2 } + 1 } { 2 - 1 } + \dfrac { 1 } { \sqrt{ 2 } + 1 }$
$\dfrac { \sqrt{ 2 } + 1 } { \color{#FF6800}{ 2 } \color{#FF6800}{ - } \color{#FF6800}{ 1 } } + \dfrac { 1 } { \sqrt{ 2 } + 1 }$
 Subtract $1$ from $2$
$\dfrac { \sqrt{ 2 } + 1 } { \color{#FF6800}{ 1 } } + \dfrac { 1 } { \sqrt{ 2 } + 1 }$
$\dfrac { \sqrt{ 2 } + 1 } { \color{#FF6800}{ 1 } } + \dfrac { 1 } { \sqrt{ 2 } + 1 }$
 If the denominator is 1, the denominator can be removed 
$\sqrt{ \color{#FF6800}{ 2 } } + \color{#FF6800}{ 1 } + \dfrac { 1 } { \sqrt{ 2 } + 1 }$
$\sqrt{ 2 } + 1 + \dfrac { 1 } { \sqrt{ 2 } + 1 }$
 Find the conjugate irrational number of denominator 
$\sqrt{ 2 } + 1 + \color{#FF6800}{ \dfrac { 1 } { \sqrt{ 2 } + 1 } } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { \sqrt{ 2 } - 1 } { \sqrt{ 2 } - 1 } }$
$\sqrt{ 2 } + 1 + \dfrac { 1 } { \sqrt{ 2 } + 1 } \times \dfrac { \sqrt{ 2 } - 1 } { \sqrt{ 2 } - 1 }$
 The denominator is multiplied by denominator, and the numerator is multiplied by numerator 
$\sqrt{ 2 } + 1 + \color{#FF6800}{ \dfrac { 1 \left ( \sqrt{ 2 } - 1 \right ) } { \left ( \sqrt{ 2 } + 1 \right ) \left ( \sqrt{ 2 } - 1 \right ) } }$
$\sqrt{ 2 } + 1 + \dfrac { \color{#FF6800}{ 1 } \left ( \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) } { \left ( \sqrt{ 2 } + 1 \right ) \left ( \sqrt{ 2 } - 1 \right ) }$
 Multiply each term in parentheses by $1$
$\sqrt{ 2 } + 1 + \dfrac { \color{#FF6800}{ 1 } \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 } } { \left ( \sqrt{ 2 } + 1 \right ) \left ( \sqrt{ 2 } - 1 \right ) }$
$\sqrt{ 2 } + 1 + \dfrac { 1 \sqrt{ 2 } - 1 } { \left ( \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) \left ( \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) }$
 Expand the expression using $\left(a - b\right)\left(a + b\right) = a^{2} - b^{2}$
$\sqrt{ 2 } + 1 + \dfrac { 1 \sqrt{ 2 } - 1 } { \left ( \sqrt{ \color{#FF6800}{ 2 } } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 } ^ { \color{#FF6800}{ 2 } } }$
$\sqrt{ 2 } + 1 + \dfrac { 1 \sqrt{ 2 } - 1 } { \left ( \sqrt{ \color{#FF6800}{ 2 } } \right ) ^ { \color{#FF6800}{ 2 } } - 1 ^ { 2 } }$
 Calculate power 
$\sqrt{ 2 } + 1 + \dfrac { 1 \sqrt{ 2 } - 1 } { \color{#FF6800}{ 2 } - 1 ^ { 2 } }$
$\sqrt{ 2 } + 1 + \dfrac { 1 \sqrt{ 2 } - 1 } { 2 - \color{#FF6800}{ 1 } ^ { \color{#FF6800}{ 2 } } }$
 Calculate power 
$\sqrt{ 2 } + 1 + \dfrac { 1 \sqrt{ 2 } - 1 } { 2 - \color{#FF6800}{ 1 } }$
$\sqrt{ 2 } + 1 + \dfrac { \color{#FF6800}{ 1 } \sqrt{ 2 } - 1 } { 2 - 1 }$
 Multiplying any number by 1 does not change the value 
$\sqrt{ 2 } + 1 + \dfrac { \sqrt{ 2 } - 1 } { 2 - 1 }$
$\sqrt{ 2 } + 1 + \dfrac { \sqrt{ 2 } - 1 } { \color{#FF6800}{ 2 } \color{#FF6800}{ - } \color{#FF6800}{ 1 } }$
 Subtract $1$ from $2$
$\sqrt{ 2 } + 1 + \dfrac { \sqrt{ 2 } - 1 } { \color{#FF6800}{ 1 } }$
$\sqrt{ 2 } + 1 + \dfrac { \sqrt{ 2 } - 1 } { \color{#FF6800}{ 1 } }$
 If the denominator is 1, the denominator can be removed 
$\sqrt{ 2 } + 1 + \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 }$
$\sqrt{ 2 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } + \sqrt{ 2 } \color{#FF6800}{ - } \color{#FF6800}{ 1 }$
 Remove the two numbers if the values are the same and the signs are different 
$\sqrt{ 2 } + \sqrt{ 2 }$
$\sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 2 } }$
 Calculate between similar terms 
$\color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } }$
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