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$- 2 - 2 \sqrt{ 2 }$
Calculate the value
$\dfrac { 2 } { 1 - \sqrt{ 2 } }$
$ $ Find the conjugate irrational number of denominator $ $
$\color{#FF6800}{ \dfrac { \color{#FF6800}{ 2 } } { \color{#FF6800}{ 1 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } } } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 2 } } } { \color{#FF6800}{ 1 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 2 } } } }$
$\dfrac { 2 } { 1 - \sqrt{ 2 } } \times \dfrac { 1 + \sqrt{ 2 } } { 1 + \sqrt{ 2 } }$
$ $ The denominator is multiplied by denominator, and the numerator is multiplied by numerator $ $
$\color{#FF6800}{ \dfrac { \color{#FF6800}{ 2 } \left ( \color{#FF6800}{ 1 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 2 } } \right ) } { \left ( \color{#FF6800}{ 1 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } \right ) \left ( \color{#FF6800}{ 1 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 2 } } \right ) } }$
$\dfrac { \color{#FF6800}{ 2 } \left ( \color{#FF6800}{ 1 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 2 } } \right ) } { \left ( 1 - \sqrt{ 2 } \right ) \left ( 1 + \sqrt{ 2 } \right ) }$
$ $ Multiply each term in parentheses by $ 2$
$\dfrac { \color{#FF6800}{ 2 } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } } { \left ( 1 - \sqrt{ 2 } \right ) \left ( 1 + \sqrt{ 2 } \right ) }$
$\dfrac { 2 + 2 \sqrt{ 2 } } { \left ( \color{#FF6800}{ 1 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } \right ) \left ( \color{#FF6800}{ 1 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 2 } } \right ) }$
$ $ Expand the expression using $ \left(a - b\right)\left(a + b\right) = a^{2} - b^{2}$
$\dfrac { 2 + 2 \sqrt{ 2 } } { \color{#FF6800}{ 1 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \left ( \sqrt{ \color{#FF6800}{ 2 } } \right ) ^ { \color{#FF6800}{ 2 } } }$
$\dfrac { 2 + 2 \sqrt{ 2 } } { \color{#FF6800}{ 1 } ^ { \color{#FF6800}{ 2 } } - \left ( \sqrt{ 2 } \right ) ^ { 2 } }$
$ $ Calculate power $ $
$\dfrac { 2 + 2 \sqrt{ 2 } } { \color{#FF6800}{ 1 } - \left ( \sqrt{ 2 } \right ) ^ { 2 } }$
$\dfrac { 2 + 2 \sqrt{ 2 } } { 1 - \left ( \sqrt{ \color{#FF6800}{ 2 } } \right ) ^ { \color{#FF6800}{ 2 } } }$
$ $ Calculate power $ $
$\dfrac { 2 + 2 \sqrt{ 2 } } { 1 - \color{#FF6800}{ 2 } }$
$\dfrac { 2 + 2 \sqrt{ 2 } } { \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } }$
$ $ Subtract $ 2 $ from $ 1$
$\dfrac { 2 + 2 \sqrt{ 2 } } { \color{#FF6800}{ - } \color{#FF6800}{ 1 } }$
$\dfrac { 2 + 2 \sqrt{ 2 } } { \color{#FF6800}{ - } 1 }$
$ $ If the denominator is 1, the denominator can be removed $ $
$\color{#FF6800}{ - } \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \right )$
$\color{#FF6800}{ - } \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \right )$
$ $ Change the symbol of each term in parentheses when there is a (-) symbol in front of parentheses $ $
$\color{#FF6800}{ - } \color{#FF6800}{ 2 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } }$
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