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Formula
Calculate the value
$\dfrac{ 1 }{ 2- \sqrt{ 5 } } + \dfrac{ 1 }{ 2+ \sqrt{ 5 } }$
$- 4$
Calculate the value
$\dfrac { 1 } { 2 - \sqrt{ 5 } } + \dfrac { 1 } { 2 + \sqrt{ 5 } }$
 Find the conjugate irrational number of denominator 
$\color{#FF6800}{ \dfrac { 1 } { 2 - \sqrt{ 5 } } } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 2 + \sqrt{ 5 } } { 2 + \sqrt{ 5 } } } + \dfrac { 1 } { 2 + \sqrt{ 5 } }$
$\dfrac { 1 } { 2 - \sqrt{ 5 } } \times \dfrac { 2 + \sqrt{ 5 } } { 2 + \sqrt{ 5 } } + \dfrac { 1 } { 2 + \sqrt{ 5 } }$
 The denominator is multiplied by denominator, and the numerator is multiplied by numerator 
$\color{#FF6800}{ \dfrac { 1 \left ( 2 + \sqrt{ 5 } \right ) } { \left ( 2 - \sqrt{ 5 } \right ) \left ( 2 + \sqrt{ 5 } \right ) } } + \dfrac { 1 } { 2 + \sqrt{ 5 } }$
$\dfrac { \color{#FF6800}{ 1 } \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 5 } } \right ) } { \left ( 2 - \sqrt{ 5 } \right ) \left ( 2 + \sqrt{ 5 } \right ) } + \dfrac { 1 } { 2 + \sqrt{ 5 } }$
 Multiply each term in parentheses by $1$
$\dfrac { \color{#FF6800}{ 2 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \sqrt{ \color{#FF6800}{ 5 } } } { \left ( 2 - \sqrt{ 5 } \right ) \left ( 2 + \sqrt{ 5 } \right ) } + \dfrac { 1 } { 2 + \sqrt{ 5 } }$
$\dfrac { 2 + 1 \sqrt{ 5 } } { \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } \right ) \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 5 } } \right ) } + \dfrac { 1 } { 2 + \sqrt{ 5 } }$
 Expand the expression using $\left(a - b\right)\left(a + b\right) = a^{2} - b^{2}$
$\dfrac { 2 + 1 \sqrt{ 5 } } { \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \left ( \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 2 } } } + \dfrac { 1 } { 2 + \sqrt{ 5 } }$
$\dfrac { 2 + 1 \sqrt{ 5 } } { \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } - \left ( \sqrt{ 5 } \right ) ^ { 2 } } + \dfrac { 1 } { 2 + \sqrt{ 5 } }$
 Calculate power 
$\dfrac { 2 + 1 \sqrt{ 5 } } { \color{#FF6800}{ 4 } - \left ( \sqrt{ 5 } \right ) ^ { 2 } } + \dfrac { 1 } { 2 + \sqrt{ 5 } }$
$\dfrac { 2 + 1 \sqrt{ 5 } } { 4 - \left ( \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 2 } } } + \dfrac { 1 } { 2 + \sqrt{ 5 } }$
 Calculate power 
$\dfrac { 2 + 1 \sqrt{ 5 } } { 4 - \color{#FF6800}{ 5 } } + \dfrac { 1 } { 2 + \sqrt{ 5 } }$
$\dfrac { 2 + \color{#FF6800}{ 1 } \sqrt{ 5 } } { 4 - 5 } + \dfrac { 1 } { 2 + \sqrt{ 5 } }$
 Multiplying any number by 1 does not change the value 
$\dfrac { 2 + \sqrt{ 5 } } { 4 - 5 } + \dfrac { 1 } { 2 + \sqrt{ 5 } }$
$\dfrac { 2 + \sqrt{ 5 } } { \color{#FF6800}{ 4 } \color{#FF6800}{ - } \color{#FF6800}{ 5 } } + \dfrac { 1 } { 2 + \sqrt{ 5 } }$
 Subtract $5$ from $4$
$\dfrac { 2 + \sqrt{ 5 } } { \color{#FF6800}{ - } \color{#FF6800}{ 1 } } + \dfrac { 1 } { 2 + \sqrt{ 5 } }$
$\dfrac { 2 + \sqrt{ 5 } } { \color{#FF6800}{ - } 1 } + \dfrac { 1 } { 2 + \sqrt{ 5 } }$
 If the denominator is 1, the denominator can be removed 
$\color{#FF6800}{ - } \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 5 } } \right ) + \dfrac { 1 } { 2 + \sqrt{ 5 } }$
$\color{#FF6800}{ - } \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 5 } } \right ) + \dfrac { 1 } { 2 + \sqrt{ 5 } }$
 Change the symbol of each term in parentheses when there is a (-) symbol in front of parentheses 
$\color{#FF6800}{ - } \color{#FF6800}{ 2 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } + \dfrac { 1 } { 2 + \sqrt{ 5 } }$
$- 2 - \sqrt{ 5 } + \dfrac { 1 } { 2 + \sqrt{ 5 } }$
 Find the conjugate irrational number of denominator 
$- 2 - \sqrt{ 5 } + \color{#FF6800}{ \dfrac { 1 } { 2 + \sqrt{ 5 } } } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 2 - \sqrt{ 5 } } { 2 - \sqrt{ 5 } } }$
$- 2 - \sqrt{ 5 } + \dfrac { 1 } { 2 + \sqrt{ 5 } } \times \dfrac { 2 - \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
 The denominator is multiplied by denominator, and the numerator is multiplied by numerator 
$- 2 - \sqrt{ 5 } + \color{#FF6800}{ \dfrac { 1 \left ( 2 - \sqrt{ 5 } \right ) } { \left ( 2 + \sqrt{ 5 } \right ) \left ( 2 - \sqrt{ 5 } \right ) } }$
$- 2 - \sqrt{ 5 } + \dfrac { \color{#FF6800}{ 1 } \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } \right ) } { \left ( 2 + \sqrt{ 5 } \right ) \left ( 2 - \sqrt{ 5 } \right ) }$
 Multiply each term in parentheses by $1$
$- 2 - \sqrt{ 5 } + \dfrac { \color{#FF6800}{ 2 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } \right ) } { \left ( 2 + \sqrt{ 5 } \right ) \left ( 2 - \sqrt{ 5 } \right ) }$
$- 2 - \sqrt{ 5 } + \dfrac { 2 + 1 \times \left ( - \sqrt{ 5 } \right ) } { \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 5 } } \right ) \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } \right ) }$
 Expand the expression using $\left(a - b\right)\left(a + b\right) = a^{2} - b^{2}$
$- 2 - \sqrt{ 5 } + \dfrac { 2 + 1 \times \left ( - \sqrt{ 5 } \right ) } { \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \left ( \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 2 } } }$
$- 2 - \sqrt{ 5 } + \dfrac { 2 + 1 \times \left ( - \sqrt{ 5 } \right ) } { \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } - \left ( \sqrt{ 5 } \right ) ^ { 2 } }$
 Calculate power 
$- 2 - \sqrt{ 5 } + \dfrac { 2 + 1 \times \left ( - \sqrt{ 5 } \right ) } { \color{#FF6800}{ 4 } - \left ( \sqrt{ 5 } \right ) ^ { 2 } }$
$- 2 - \sqrt{ 5 } + \dfrac { 2 + 1 \times \left ( - \sqrt{ 5 } \right ) } { 4 - \left ( \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 2 } } }$
 Calculate power 
$- 2 - \sqrt{ 5 } + \dfrac { 2 + 1 \times \left ( - \sqrt{ 5 } \right ) } { 4 - \color{#FF6800}{ 5 } }$
$- 2 - \sqrt{ 5 } + \dfrac { 2 + \color{#FF6800}{ 1 } \times \left ( - \sqrt{ 5 } \right ) } { 4 - 5 }$
 Multiplying any number by 1 does not change the value 
$- 2 - \sqrt{ 5 } + \dfrac { 2 - \sqrt{ 5 } } { 4 - 5 }$
$- 2 - \sqrt{ 5 } + \dfrac { 2 - \sqrt{ 5 } } { \color{#FF6800}{ 4 } \color{#FF6800}{ - } \color{#FF6800}{ 5 } }$
 Subtract $5$ from $4$
$- 2 - \sqrt{ 5 } + \dfrac { 2 - \sqrt{ 5 } } { \color{#FF6800}{ - } \color{#FF6800}{ 1 } }$
$- 2 - \sqrt{ 5 } + \dfrac { 2 - \sqrt{ 5 } } { \color{#FF6800}{ - } 1 }$
 If the denominator is 1, the denominator can be removed 
$- 2 - \sqrt{ 5 } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } \right )$
$- 2 - \sqrt{ 5 } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } \right )$
 Change the symbol of each term in parentheses when there is a (-) symbol in front of parentheses 
$- 2 - \sqrt{ 5 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } + \sqrt{ \color{#FF6800}{ 5 } }$
$- 2 \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } - 2 \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 5 } }$
 Remove the two numbers if the values are the same and the signs are different 
$- 2 - 2$
$\color{#FF6800}{ - } \color{#FF6800}{ 2 } \color{#FF6800}{ - } \color{#FF6800}{ 2 }$
 Find the sum of the negative numbers 
$\color{#FF6800}{ - } \color{#FF6800}{ 4 }$
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