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Calculate the value
$\dfrac{ 1 }{ 1+ \sqrt{ 3 } }$
$- \dfrac { 1 - \sqrt{ 3 } } { 2 }$
Calculate the value
$\dfrac { 1 } { 1 + \sqrt{ 3 } }$
 Find the conjugate irrational number of denominator 
$\color{#FF6800}{ \dfrac { 1 } { 1 + \sqrt{ 3 } } } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 1 - \sqrt{ 3 } } { 1 - \sqrt{ 3 } } }$
$\dfrac { 1 } { 1 + \sqrt{ 3 } } \times \dfrac { 1 - \sqrt{ 3 } } { 1 - \sqrt{ 3 } }$
 The denominator is multiplied by denominator, and the numerator is multiplied by numerator 
$\color{#FF6800}{ \dfrac { 1 \left ( 1 - \sqrt{ 3 } \right ) } { \left ( 1 + \sqrt{ 3 } \right ) \left ( 1 - \sqrt{ 3 } \right ) } }$
$\dfrac { \color{#FF6800}{ 1 } \left ( \color{#FF6800}{ 1 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 3 } } \right ) } { \left ( 1 + \sqrt{ 3 } \right ) \left ( 1 - \sqrt{ 3 } \right ) }$
 Multiply each term in parentheses by $1$
$\dfrac { \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 3 } } \right ) } { \left ( 1 + \sqrt{ 3 } \right ) \left ( 1 - \sqrt{ 3 } \right ) }$
$\dfrac { 1 + 1 \times \left ( - \sqrt{ 3 } \right ) } { \left ( \color{#FF6800}{ 1 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 3 } } \right ) \left ( \color{#FF6800}{ 1 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 3 } } \right ) }$
 Expand the expression using $\left(a - b\right)\left(a + b\right) = a^{2} - b^{2}$
$\dfrac { 1 + 1 \times \left ( - \sqrt{ 3 } \right ) } { \color{#FF6800}{ 1 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \left ( \sqrt{ \color{#FF6800}{ 3 } } \right ) ^ { \color{#FF6800}{ 2 } } }$
$\dfrac { 1 + 1 \times \left ( - \sqrt{ 3 } \right ) } { \color{#FF6800}{ 1 } ^ { \color{#FF6800}{ 2 } } - \left ( \sqrt{ 3 } \right ) ^ { 2 } }$
 Calculate power 
$\dfrac { 1 + 1 \times \left ( - \sqrt{ 3 } \right ) } { \color{#FF6800}{ 1 } - \left ( \sqrt{ 3 } \right ) ^ { 2 } }$
$\dfrac { 1 + 1 \times \left ( - \sqrt{ 3 } \right ) } { 1 - \left ( \sqrt{ \color{#FF6800}{ 3 } } \right ) ^ { \color{#FF6800}{ 2 } } }$
 Calculate power 
$\dfrac { 1 + 1 \times \left ( - \sqrt{ 3 } \right ) } { 1 - \color{#FF6800}{ 3 } }$
$\dfrac { 1 + \color{#FF6800}{ 1 } \times \left ( - \sqrt{ 3 } \right ) } { 1 - 3 }$
 Multiplying any number by 1 does not change the value 
$\dfrac { 1 - \sqrt{ 3 } } { 1 - 3 }$
$\dfrac { 1 - \sqrt{ 3 } } { \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ 3 } }$
 Subtract $3$ from $1$
$\dfrac { 1 - \sqrt{ 3 } } { \color{#FF6800}{ - } \color{#FF6800}{ 2 } }$
$\dfrac { 1 - \sqrt{ 3 } } { \color{#FF6800}{ - } \color{#FF6800}{ 2 } }$
 Move the minus sign to the front of the fraction 
$\color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 - \sqrt{ 3 } } { 2 } }$
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