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$\dfrac{ \sqrt{ 3 } }{ 2 \sqrt{ 3 } -3 }$
$2 + \sqrt{ 3 }$
Calculate the value
$\dfrac { \sqrt{ 3 } } { 2 \sqrt{ 3 } - 3 }$
$ $ Find the conjugate irrational number of denominator $ $
$\color{#FF6800}{ \dfrac { \sqrt{ 3 } } { 2 \sqrt{ 3 } - 3 } } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 2 \sqrt{ 3 } + 3 } { 2 \sqrt{ 3 } + 3 } }$
$\dfrac { \sqrt{ 3 } } { 2 \sqrt{ 3 } - 3 } \times \dfrac { 2 \sqrt{ 3 } + 3 } { 2 \sqrt{ 3 } + 3 }$
$ $ The denominator is multiplied by denominator, and the numerator is multiplied by numerator $ $
$\color{#FF6800}{ \dfrac { \sqrt{ 3 } \left ( 2 \sqrt{ 3 } + 3 \right ) } { \left ( 2 \sqrt{ 3 } - 3 \right ) \left ( 2 \sqrt{ 3 } + 3 \right ) } }$
$\dfrac { \sqrt{ \color{#FF6800}{ 3 } } \left ( \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \right ) } { \left ( 2 \sqrt{ 3 } - 3 \right ) \left ( 2 \sqrt{ 3 } + 3 \right ) }$
$ $ Multiply each term in parentheses by $ \sqrt{ 3 }$
$\dfrac { \sqrt{ \color{#FF6800}{ 3 } } \left ( \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 3 } } \right ) \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } } { \left ( 2 \sqrt{ 3 } - 3 \right ) \left ( 2 \sqrt{ 3 } + 3 \right ) }$
$\dfrac { \sqrt{ 3 } \left ( 2 \sqrt{ 3 } \right ) + \sqrt{ 3 } \times 3 } { \left ( \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \right ) \left ( \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \right ) }$
$ $ Expand the expression using $ \left(a - b\right)\left(a + b\right) = a^{2} - b^{2}$
$\dfrac { \sqrt{ 3 } \left ( 2 \sqrt{ 3 } \right ) + \sqrt{ 3 } \times 3 } { \left ( \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 3 } } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } } }$
$\dfrac { \sqrt{ 3 } \left ( 2 \sqrt{ 3 } \right ) + \sqrt{ 3 } \times 3 } { \left ( \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 3 } } \right ) ^ { \color{#FF6800}{ 2 } } - 3 ^ { 2 } }$
$ $ Calculate power $ $
$\dfrac { \sqrt{ 3 } \left ( 2 \sqrt{ 3 } \right ) + \sqrt{ 3 } \times 3 } { \color{#FF6800}{ 12 } - 3 ^ { 2 } }$
$\dfrac { \sqrt{ 3 } \left ( 2 \sqrt{ 3 } \right ) + \sqrt{ 3 } \times 3 } { 12 - \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } } }$
$ $ Calculate power $ $
$\dfrac { \sqrt{ 3 } \left ( 2 \sqrt{ 3 } \right ) + \sqrt{ 3 } \times 3 } { 12 - \color{#FF6800}{ 9 } }$
$\dfrac { \sqrt{ \color{#FF6800}{ 3 } } \left ( \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 3 } } \right ) + \sqrt{ 3 } \times 3 } { 12 - 9 }$
$ $ Get rid of unnecessary parentheses $ $
$\dfrac { \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 3 } } + \sqrt{ 3 } \times 3 } { 12 - 9 }$
$\dfrac { \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 3 } } + \sqrt{ 3 } \times 3 } { 12 - 9 }$
$ $ Simplify the expression $ $
$\dfrac { \color{#FF6800}{ 6 } + \sqrt{ 3 } \times 3 } { 12 - 9 }$
$\dfrac { 6 + \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } } { 12 - 9 }$
$ $ Simplify the expression $ $
$\dfrac { 6 + \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 3 } } } { 12 - 9 }$
$\dfrac { 6 + 3 \sqrt{ 3 } } { \color{#FF6800}{ 12 } \color{#FF6800}{ - } \color{#FF6800}{ 9 } }$
$ $ Subtract $ 9 $ from $ 12$
$\dfrac { 6 + 3 \sqrt{ 3 } } { \color{#FF6800}{ 3 } }$
$\color{#FF6800}{ \dfrac { 6 + 3 \sqrt{ 3 } } { 3 } }$
$ $ Reduce the fraction $ $
$\color{#FF6800}{ 2 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 3 } }$
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