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$\dfrac{ 1 }{ 8+3 \sqrt{ 7 } }$
$8 - 3 \sqrt{ 7 }$
Calculate the value
$\dfrac { 1 } { 8 + 3 \sqrt{ 7 } }$
$ $ Find the conjugate irrational number of denominator $ $
$\color{#FF6800}{ \dfrac { 1 } { 8 + 3 \sqrt{ 7 } } } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 8 - \left ( 3 \sqrt{ 7 } \right ) } { 8 - \left ( 3 \sqrt{ 7 } \right ) } }$
$\dfrac { 1 } { 8 + 3 \sqrt{ 7 } } \times \dfrac { 8 - \left ( 3 \sqrt{ 7 } \right ) } { 8 - \left ( 3 \sqrt{ 7 } \right ) }$
$ $ The denominator is multiplied by denominator, and the numerator is multiplied by numerator $ $
$\color{#FF6800}{ \dfrac { 1 \left ( 8 - \left ( 3 \sqrt{ 7 } \right ) \right ) } { \left ( 8 + 3 \sqrt{ 7 } \right ) \left ( 8 - \left ( 3 \sqrt{ 7 } \right ) \right ) } }$
$\dfrac { \color{#FF6800}{ 1 } \left ( \color{#FF6800}{ 8 } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 7 } } \right ) \right ) } { \left ( 8 + 3 \sqrt{ 7 } \right ) \left ( 8 - \left ( 3 \sqrt{ 7 } \right ) \right ) }$
$ $ Multiply each term in parentheses by $ 1$
$\dfrac { \color{#FF6800}{ 8 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \left ( \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 7 } } \right ) \right ) } { \left ( 8 + 3 \sqrt{ 7 } \right ) \left ( 8 - \left ( 3 \sqrt{ 7 } \right ) \right ) }$
$\dfrac { 8 + 1 \times \left ( - \left ( 3 \sqrt{ 7 } \right ) \right ) } { \left ( \color{#FF6800}{ 8 } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 7 } } \right ) \left ( \color{#FF6800}{ 8 } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 7 } } \right ) \right ) }$
$ $ Expand the expression using $ \left(a - b\right)\left(a + b\right) = a^{2} - b^{2}$
$\dfrac { 8 + 1 \times \left ( - \left ( 3 \sqrt{ 7 } \right ) \right ) } { \color{#FF6800}{ 8 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 7 } } \right ) ^ { \color{#FF6800}{ 2 } } }$
$\dfrac { 8 + 1 \times \left ( - \left ( 3 \sqrt{ 7 } \right ) \right ) } { \color{#FF6800}{ 8 } ^ { \color{#FF6800}{ 2 } } - \left ( 3 \sqrt{ 7 } \right ) ^ { 2 } }$
$ $ Calculate power $ $
$\dfrac { 8 + 1 \times \left ( - \left ( 3 \sqrt{ 7 } \right ) \right ) } { \color{#FF6800}{ 64 } - \left ( 3 \sqrt{ 7 } \right ) ^ { 2 } }$
$\dfrac { 8 + 1 \times \left ( - \left ( 3 \sqrt{ 7 } \right ) \right ) } { 64 - \left ( \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 7 } } \right ) ^ { \color{#FF6800}{ 2 } } }$
$ $ Calculate power $ $
$\dfrac { 8 + 1 \times \left ( - \left ( 3 \sqrt{ 7 } \right ) \right ) } { 64 - \color{#FF6800}{ 63 } }$
$\dfrac { 8 + \color{#FF6800}{ 1 } \times \left ( - \left ( 3 \sqrt{ 7 } \right ) \right ) } { 64 - 63 }$
$ $ Multiplying any number by 1 does not change the value $ $
$\dfrac { 8 - \left ( 3 \sqrt{ 7 } \right ) } { 64 - 63 }$
$\dfrac { 8 - \left ( 3 \sqrt{ 7 } \right ) } { \color{#FF6800}{ 64 } \color{#FF6800}{ - } \color{#FF6800}{ 63 } }$
$ $ Subtract $ 63 $ from $ 64$
$\dfrac { 8 - \left ( 3 \sqrt{ 7 } \right ) } { \color{#FF6800}{ 1 } }$
$\dfrac { 8 - \left ( 3 \sqrt{ 7 } \right ) } { \color{#FF6800}{ 1 } }$
$ $ If the denominator is 1, the denominator can be removed $ $
$\color{#FF6800}{ 8 } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 7 } } \right )$
$8 \color{#FF6800}{ - } \left ( \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 7 } } \right )$
$ $ Get rid of unnecessary parentheses $ $
$8 \color{#FF6800}{ - } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 7 } }$
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