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$-3+ \sqrt{ 10 } - \dfrac{ 1 }{ -3+ \sqrt{ 10 } }$
$- 6$
Calculate the value
$- 3 + \sqrt{ 10 } - \dfrac { 1 } { - 3 + \sqrt{ 10 } }$
$ $ Change the position of the left and right terms to make them easier to rationalize $ $
$- 3 + \sqrt{ 10 } - \dfrac { 1 } { \sqrt{ \color{#FF6800}{ 10 } } \color{#FF6800}{ - } \color{#FF6800}{ 3 } }$
$- 3 + \sqrt{ 10 } - \dfrac { 1 } { \sqrt{ 10 } - 3 }$
$ $ Find the conjugate irrational number of denominator $ $
$- 3 + \sqrt{ 10 } - \left ( \color{#FF6800}{ \dfrac { 1 } { \sqrt{ 10 } - 3 } } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { \sqrt{ 10 } + 3 } { \sqrt{ 10 } + 3 } } \right )$
$- 3 + \sqrt{ 10 } - \left ( \dfrac { 1 } { \sqrt{ 10 } - 3 } \times \dfrac { \sqrt{ 10 } + 3 } { \sqrt{ 10 } + 3 } \right )$
$ $ The denominator is multiplied by denominator, and the numerator is multiplied by numerator $ $
$- 3 + \sqrt{ 10 } - \color{#FF6800}{ \dfrac { 1 \left ( \sqrt{ 10 } + 3 \right ) } { \left ( \sqrt{ 10 } - 3 \right ) \left ( \sqrt{ 10 } + 3 \right ) } }$
$- 3 + \sqrt{ 10 } - \dfrac { \color{#FF6800}{ 1 } \left ( \sqrt{ \color{#FF6800}{ 10 } } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \right ) } { \left ( \sqrt{ 10 } - 3 \right ) \left ( \sqrt{ 10 } + 3 \right ) }$
$ $ Multiply each term in parentheses by $ 1$
$- 3 + \sqrt{ 10 } - \dfrac { \color{#FF6800}{ 1 } \sqrt{ \color{#FF6800}{ 10 } } \color{#FF6800}{ + } \color{#FF6800}{ 3 } } { \left ( \sqrt{ 10 } - 3 \right ) \left ( \sqrt{ 10 } + 3 \right ) }$
$- 3 + \sqrt{ 10 } - \dfrac { 1 \sqrt{ 10 } + 3 } { \left ( \sqrt{ \color{#FF6800}{ 10 } } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \right ) \left ( \sqrt{ \color{#FF6800}{ 10 } } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \right ) }$
$ $ Expand the expression using $ \left(a - b\right)\left(a + b\right) = a^{2} - b^{2}$
$- 3 + \sqrt{ 10 } - \dfrac { 1 \sqrt{ 10 } + 3 } { \left ( \sqrt{ \color{#FF6800}{ 10 } } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } } }$
$- 3 + \sqrt{ 10 } - \dfrac { 1 \sqrt{ 10 } + 3 } { \left ( \sqrt{ \color{#FF6800}{ 10 } } \right ) ^ { \color{#FF6800}{ 2 } } - 3 ^ { 2 } }$
$ $ Calculate power $ $
$- 3 + \sqrt{ 10 } - \dfrac { 1 \sqrt{ 10 } + 3 } { \color{#FF6800}{ 10 } - 3 ^ { 2 } }$
$- 3 + \sqrt{ 10 } - \dfrac { 1 \sqrt{ 10 } + 3 } { 10 - \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } } }$
$ $ Calculate power $ $
$- 3 + \sqrt{ 10 } - \dfrac { 1 \sqrt{ 10 } + 3 } { 10 - \color{#FF6800}{ 9 } }$
$- 3 + \sqrt{ 10 } - \dfrac { \color{#FF6800}{ 1 } \sqrt{ 10 } + 3 } { 10 - 9 }$
$ $ Multiplying any number by 1 does not change the value $ $
$- 3 + \sqrt{ 10 } - \dfrac { \sqrt{ 10 } + 3 } { 10 - 9 }$
$- 3 + \sqrt{ 10 } - \dfrac { \sqrt{ 10 } + 3 } { \color{#FF6800}{ 10 } \color{#FF6800}{ - } \color{#FF6800}{ 9 } }$
$ $ Subtract $ 9 $ from $ 10$
$- 3 + \sqrt{ 10 } - \dfrac { \sqrt{ 10 } + 3 } { \color{#FF6800}{ 1 } }$
$- 3 + \sqrt{ 10 } - \dfrac { \sqrt{ 10 } + 3 } { \color{#FF6800}{ 1 } }$
$ $ If the denominator is 1, the denominator can be removed $ $
$- 3 + \sqrt{ 10 } - \left ( \sqrt{ \color{#FF6800}{ 10 } } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \right )$
$- 3 + \sqrt{ 10 } \color{#FF6800}{ - } \left ( \sqrt{ \color{#FF6800}{ 10 } } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \right )$
$ $ Change the symbol of each term in parentheses when there is a (-) symbol in front of parentheses $ $
$- 3 + \sqrt{ 10 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 10 } } \color{#FF6800}{ - } \color{#FF6800}{ 3 }$
$- 3 \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 10 } } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 10 } } - 3$
$ $ Remove the two numbers if the values are the same and the signs are different $ $
$- 3 - 3$
$\color{#FF6800}{ - } \color{#FF6800}{ 3 } \color{#FF6800}{ - } \color{#FF6800}{ 3 }$
$ $ Find the sum of the negative numbers $ $
$\color{#FF6800}{ - } \color{#FF6800}{ 6 }$
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