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$\dfrac{ 4 }{ 3+ \sqrt{ 5 } }$
$3 - \sqrt{ 5 }$
Calculate the value
$\dfrac { 4 } { 3 + \sqrt{ 5 } }$
$ $ Find the conjugate irrational number of denominator $ $
$\color{#FF6800}{ \dfrac { 4 } { 3 + \sqrt{ 5 } } } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 3 - \sqrt{ 5 } } { 3 - \sqrt{ 5 } } }$
$\dfrac { 4 } { 3 + \sqrt{ 5 } } \times \dfrac { 3 - \sqrt{ 5 } } { 3 - \sqrt{ 5 } }$
$ $ The denominator is multiplied by denominator, and the numerator is multiplied by numerator $ $
$\color{#FF6800}{ \dfrac { 4 \left ( 3 - \sqrt{ 5 } \right ) } { \left ( 3 + \sqrt{ 5 } \right ) \left ( 3 - \sqrt{ 5 } \right ) } }$
$\dfrac { \color{#FF6800}{ 4 } \left ( \color{#FF6800}{ 3 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } \right ) } { \left ( 3 + \sqrt{ 5 } \right ) \left ( 3 - \sqrt{ 5 } \right ) }$
$ $ Multiply each term in parentheses by $ 4$
$\dfrac { \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } \right ) } { \left ( 3 + \sqrt{ 5 } \right ) \left ( 3 - \sqrt{ 5 } \right ) }$
$\dfrac { 4 \times 3 + 4 \times \left ( - \sqrt{ 5 } \right ) } { \left ( \color{#FF6800}{ 3 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 5 } } \right ) \left ( \color{#FF6800}{ 3 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } \right ) }$
$ $ Expand the expression using $ \left(a - b\right)\left(a + b\right) = a^{2} - b^{2}$
$\dfrac { 4 \times 3 + 4 \times \left ( - \sqrt{ 5 } \right ) } { \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \left ( \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 2 } } }$
$\dfrac { 4 \times 3 + 4 \times \left ( - \sqrt{ 5 } \right ) } { \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } } - \left ( \sqrt{ 5 } \right ) ^ { 2 } }$
$ $ Calculate power $ $
$\dfrac { 4 \times 3 + 4 \times \left ( - \sqrt{ 5 } \right ) } { \color{#FF6800}{ 9 } - \left ( \sqrt{ 5 } \right ) ^ { 2 } }$
$\dfrac { 4 \times 3 + 4 \times \left ( - \sqrt{ 5 } \right ) } { 9 - \left ( \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 2 } } }$
$ $ Calculate power $ $
$\dfrac { 4 \times 3 + 4 \times \left ( - \sqrt{ 5 } \right ) } { 9 - \color{#FF6800}{ 5 } }$
$\dfrac { \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } + 4 \times \left ( - \sqrt{ 5 } \right ) } { 9 - 5 }$
$ $ Multiply $ 4 $ and $ 3$
$\dfrac { \color{#FF6800}{ 12 } + 4 \times \left ( - \sqrt{ 5 } \right ) } { 9 - 5 }$
$\dfrac { 12 + \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } \right ) } { 9 - 5 }$
$ $ Simplify the expression $ $
$\dfrac { 12 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 5 } } } { 9 - 5 }$
$\dfrac { 12 - 4 \sqrt{ 5 } } { \color{#FF6800}{ 9 } \color{#FF6800}{ - } \color{#FF6800}{ 5 } }$
$ $ Subtract $ 5 $ from $ 9$
$\dfrac { 12 - 4 \sqrt{ 5 } } { \color{#FF6800}{ 4 } }$
$\color{#FF6800}{ \dfrac { 12 - 4 \sqrt{ 5 } } { 4 } }$
$ $ Reduce the fraction $ $
$\color{#FF6800}{ 3 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } }$
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