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$\dfrac{ 3 }{ 6+3 \sqrt{ 3 } }$
$2 - \sqrt{ 3 }$
Calculate the value
$\dfrac { 3 } { 6 + 3 \sqrt{ 3 } }$
$ $ Find the conjugate irrational number of denominator $ $
$\color{#FF6800}{ \dfrac { 3 } { 6 + 3 \sqrt{ 3 } } } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 6 - \left ( 3 \sqrt{ 3 } \right ) } { 6 - \left ( 3 \sqrt{ 3 } \right ) } }$
$\dfrac { 3 } { 6 + 3 \sqrt{ 3 } } \times \dfrac { 6 - \left ( 3 \sqrt{ 3 } \right ) } { 6 - \left ( 3 \sqrt{ 3 } \right ) }$
$ $ The denominator is multiplied by denominator, and the numerator is multiplied by numerator $ $
$\color{#FF6800}{ \dfrac { 3 \left ( 6 - \left ( 3 \sqrt{ 3 } \right ) \right ) } { \left ( 6 + 3 \sqrt{ 3 } \right ) \left ( 6 - \left ( 3 \sqrt{ 3 } \right ) \right ) } }$
$\dfrac { \color{#FF6800}{ 3 } \left ( \color{#FF6800}{ 6 } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 3 } } \right ) \right ) } { \left ( 6 + 3 \sqrt{ 3 } \right ) \left ( 6 - \left ( 3 \sqrt{ 3 } \right ) \right ) }$
$ $ Multiply each term in parentheses by $ 3$
$\dfrac { \color{#FF6800}{ 3 } \color{#FF6800}{ \times } \color{#FF6800}{ 6 } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \left ( \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 3 } } \right ) \right ) } { \left ( 6 + 3 \sqrt{ 3 } \right ) \left ( 6 - \left ( 3 \sqrt{ 3 } \right ) \right ) }$
$\dfrac { 3 \times 6 + 3 \times \left ( - \left ( 3 \sqrt{ 3 } \right ) \right ) } { \left ( \color{#FF6800}{ 6 } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 3 } } \right ) \left ( \color{#FF6800}{ 6 } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 3 } } \right ) \right ) }$
$ $ Expand the expression using $ \left(a - b\right)\left(a + b\right) = a^{2} - b^{2}$
$\dfrac { 3 \times 6 + 3 \times \left ( - \left ( 3 \sqrt{ 3 } \right ) \right ) } { \color{#FF6800}{ 6 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 3 } } \right ) ^ { \color{#FF6800}{ 2 } } }$
$\dfrac { 3 \times 6 + 3 \times \left ( - \left ( 3 \sqrt{ 3 } \right ) \right ) } { \color{#FF6800}{ 6 } ^ { \color{#FF6800}{ 2 } } - \left ( 3 \sqrt{ 3 } \right ) ^ { 2 } }$
$ $ Calculate power $ $
$\dfrac { 3 \times 6 + 3 \times \left ( - \left ( 3 \sqrt{ 3 } \right ) \right ) } { \color{#FF6800}{ 36 } - \left ( 3 \sqrt{ 3 } \right ) ^ { 2 } }$
$\dfrac { 3 \times 6 + 3 \times \left ( - \left ( 3 \sqrt{ 3 } \right ) \right ) } { 36 - \left ( \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 3 } } \right ) ^ { \color{#FF6800}{ 2 } } }$
$ $ Calculate power $ $
$\dfrac { 3 \times 6 + 3 \times \left ( - \left ( 3 \sqrt{ 3 } \right ) \right ) } { 36 - \color{#FF6800}{ 27 } }$
$\dfrac { \color{#FF6800}{ 3 } \color{#FF6800}{ \times } \color{#FF6800}{ 6 } + 3 \times \left ( - \left ( 3 \sqrt{ 3 } \right ) \right ) } { 36 - 27 }$
$ $ Multiply $ 3 $ and $ 6$
$\dfrac { \color{#FF6800}{ 18 } + 3 \times \left ( - \left ( 3 \sqrt{ 3 } \right ) \right ) } { 36 - 27 }$
$\dfrac { 18 + 3 \times \left ( \color{#FF6800}{ - } \left ( 3 \sqrt{ 3 } \right ) \right ) } { 36 - 27 }$
$ $ Move the (-) sign forward $ $
$\dfrac { 18 \color{#FF6800}{ - } 3 \left ( 3 \sqrt{ 3 } \right ) } { 36 - 27 }$
$\dfrac { 18 \color{#FF6800}{ - } \color{#FF6800}{ 3 } \left ( \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 3 } } \right ) } { 36 - 27 }$
$ $ Get rid of unnecessary parentheses $ $
$\dfrac { 18 \color{#FF6800}{ - } \color{#FF6800}{ 3 } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 3 } } } { 36 - 27 }$
$\dfrac { 18 \color{#FF6800}{ - } \color{#FF6800}{ 3 } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 3 } } } { 36 - 27 }$
$ $ Simplify the expression $ $
$\dfrac { 18 \color{#FF6800}{ - } \color{#FF6800}{ 9 } \sqrt{ \color{#FF6800}{ 3 } } } { 36 - 27 }$
$\dfrac { 18 - 9 \sqrt{ 3 } } { \color{#FF6800}{ 36 } \color{#FF6800}{ - } \color{#FF6800}{ 27 } }$
$ $ Subtract $ 27 $ from $ 36$
$\dfrac { 18 - 9 \sqrt{ 3 } } { \color{#FF6800}{ 9 } }$
$\color{#FF6800}{ \dfrac { 18 - 9 \sqrt{ 3 } } { 9 } }$
$ $ Reduce the fraction $ $
$\color{#FF6800}{ 2 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 3 } }$
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