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Calculate the value
$\dfrac{ 1 }{ 4 \sqrt{ 2 } - \sqrt{ 31 } } - \sqrt{ 44-13 }$
$4 \sqrt{ 2 }$
Calculate the value
$\dfrac { 1 } { 4 \sqrt{ 2 } - \sqrt{ 31 } } - \sqrt{ 44 - 13 }$
 Find the conjugate irrational number of denominator 
$\color{#FF6800}{ \dfrac { 1 } { 4 \sqrt{ 2 } - \sqrt{ 31 } } } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 4 \sqrt{ 2 } + \sqrt{ 31 } } { 4 \sqrt{ 2 } + \sqrt{ 31 } } } - \sqrt{ 44 - 13 }$
$\dfrac { 1 } { 4 \sqrt{ 2 } - \sqrt{ 31 } } \times \dfrac { 4 \sqrt{ 2 } + \sqrt{ 31 } } { 4 \sqrt{ 2 } + \sqrt{ 31 } } - \sqrt{ 44 - 13 }$
 The denominator is multiplied by denominator, and the numerator is multiplied by numerator 
$\color{#FF6800}{ \dfrac { 1 \left ( 4 \sqrt{ 2 } + \sqrt{ 31 } \right ) } { \left ( 4 \sqrt{ 2 } - \sqrt{ 31 } \right ) \left ( 4 \sqrt{ 2 } + \sqrt{ 31 } \right ) } } - \sqrt{ 44 - 13 }$
$\dfrac { \color{#FF6800}{ 1 } \left ( \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 31 } } \right ) } { \left ( 4 \sqrt{ 2 } - \sqrt{ 31 } \right ) \left ( 4 \sqrt{ 2 } + \sqrt{ 31 } \right ) } - \sqrt{ 44 - 13 }$
 Multiply each term in parentheses by $1$
$\dfrac { \color{#FF6800}{ 1 } \left ( \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 2 } } \right ) \color{#FF6800}{ + } \color{#FF6800}{ 1 } \sqrt{ \color{#FF6800}{ 31 } } } { \left ( 4 \sqrt{ 2 } - \sqrt{ 31 } \right ) \left ( 4 \sqrt{ 2 } + \sqrt{ 31 } \right ) } - \sqrt{ 44 - 13 }$
$\dfrac { 1 \left ( 4 \sqrt{ 2 } \right ) + 1 \sqrt{ 31 } } { \left ( \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 31 } } \right ) \left ( \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 31 } } \right ) } - \sqrt{ 44 - 13 }$
 Expand the expression using $\left(a - b\right)\left(a + b\right) = a^{2} - b^{2}$
$\dfrac { 1 \left ( 4 \sqrt{ 2 } \right ) + 1 \sqrt{ 31 } } { \left ( \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 2 } } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \left ( \sqrt{ \color{#FF6800}{ 31 } } \right ) ^ { \color{#FF6800}{ 2 } } } - \sqrt{ 44 - 13 }$
$\dfrac { 1 \left ( 4 \sqrt{ 2 } \right ) + 1 \sqrt{ 31 } } { \left ( \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 2 } } \right ) ^ { \color{#FF6800}{ 2 } } - \left ( \sqrt{ 31 } \right ) ^ { 2 } } - \sqrt{ 44 - 13 }$
 Calculate power 
$\dfrac { 1 \left ( 4 \sqrt{ 2 } \right ) + 1 \sqrt{ 31 } } { \color{#FF6800}{ 32 } - \left ( \sqrt{ 31 } \right ) ^ { 2 } } - \sqrt{ 44 - 13 }$
$\dfrac { 1 \left ( 4 \sqrt{ 2 } \right ) + 1 \sqrt{ 31 } } { 32 - \left ( \sqrt{ \color{#FF6800}{ 31 } } \right ) ^ { \color{#FF6800}{ 2 } } } - \sqrt{ 44 - 13 }$
 Calculate power 
$\dfrac { 1 \left ( 4 \sqrt{ 2 } \right ) + 1 \sqrt{ 31 } } { 32 - \color{#FF6800}{ 31 } } - \sqrt{ 44 - 13 }$
$\dfrac { \color{#FF6800}{ 1 } \left ( \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 2 } } \right ) + 1 \sqrt{ 31 } } { 32 - 31 } - \sqrt{ 44 - 13 }$
 Get rid of unnecessary parentheses 
$\dfrac { \color{#FF6800}{ 1 } \color{#FF6800}{ \times } \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 2 } } + 1 \sqrt{ 31 } } { 32 - 31 } - \sqrt{ 44 - 13 }$
$\dfrac { \color{#FF6800}{ 1 } \times 4 \sqrt{ 2 } + 1 \sqrt{ 31 } } { 32 - 31 } - \sqrt{ 44 - 13 }$
 Multiplying any number by 1 does not change the value 
$\dfrac { 4 \sqrt{ 2 } + 1 \sqrt{ 31 } } { 32 - 31 } - \sqrt{ 44 - 13 }$
$\dfrac { 4 \sqrt{ 2 } + \color{#FF6800}{ 1 } \sqrt{ 31 } } { 32 - 31 } - \sqrt{ 44 - 13 }$
 Multiplying any number by 1 does not change the value 
$\dfrac { 4 \sqrt{ 2 } + \sqrt{ 31 } } { 32 - 31 } - \sqrt{ 44 - 13 }$
$\dfrac { 4 \sqrt{ 2 } + \sqrt{ 31 } } { \color{#FF6800}{ 32 } \color{#FF6800}{ - } \color{#FF6800}{ 31 } } - \sqrt{ 44 - 13 }$
 Subtract $31$ from $32$
$\dfrac { 4 \sqrt{ 2 } + \sqrt{ 31 } } { \color{#FF6800}{ 1 } } - \sqrt{ 44 - 13 }$
$\dfrac { 4 \sqrt{ 2 } + \sqrt{ 31 } } { \color{#FF6800}{ 1 } } - \sqrt{ 44 - 13 }$
 If the denominator is 1, the denominator can be removed 
$\color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 2 } } + \sqrt{ \color{#FF6800}{ 31 } } - \sqrt{ 44 - 13 }$
$4 \sqrt{ 2 } + \sqrt{ 31 } - \sqrt{ \color{#FF6800}{ 44 } \color{#FF6800}{ - } \color{#FF6800}{ 13 } }$
 Subtract $13$ from $44$
$4 \sqrt{ 2 } + \sqrt{ 31 } - \sqrt{ \color{#FF6800}{ 31 } }$
$4 \sqrt{ 2 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 31 } } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 31 } }$
 Remove the two numbers if the values are the same and the signs are different 
$4 \sqrt{ 2 }$
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