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Calculate the value
$\dfrac{ 4 \sqrt{ 3 } -3 \sqrt{ 5 } }{ 4 \sqrt{ 3 } +3 \sqrt{ 5 } }$
$31 - 8 \sqrt{ 15 }$
Calculate the value
$\dfrac { 4 \sqrt{ 3 } - 3 \sqrt{ 5 } } { 4 \sqrt{ 3 } + 3 \sqrt{ 5 } }$
 Find the conjugate irrational number of denominator 
$\color{#FF6800}{ \dfrac { 4 \sqrt{ 3 } - 3 \sqrt{ 5 } } { 4 \sqrt{ 3 } + 3 \sqrt{ 5 } } } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 4 \sqrt{ 3 } - \left ( 3 \sqrt{ 5 } \right ) } { 4 \sqrt{ 3 } - \left ( 3 \sqrt{ 5 } \right ) } }$
$\dfrac { 4 \sqrt{ 3 } - 3 \sqrt{ 5 } } { 4 \sqrt{ 3 } + 3 \sqrt{ 5 } } \times \dfrac { 4 \sqrt{ 3 } - \left ( 3 \sqrt{ 5 } \right ) } { 4 \sqrt{ 3 } - \left ( 3 \sqrt{ 5 } \right ) }$
 The denominator is multiplied by denominator, and the numerator is multiplied by numerator 
$\color{#FF6800}{ \dfrac { \left ( 4 \sqrt{ 3 } - 3 \sqrt{ 5 } \right ) \left ( 4 \sqrt{ 3 } - \left ( 3 \sqrt{ 5 } \right ) \right ) } { \left ( 4 \sqrt{ 3 } + 3 \sqrt{ 5 } \right ) \left ( 4 \sqrt{ 3 } - \left ( 3 \sqrt{ 5 } \right ) \right ) } }$
$\dfrac { \left ( \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 5 } } \right ) \left ( \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 5 } } \right ) \right ) } { \left ( 4 \sqrt{ 3 } + 3 \sqrt{ 5 } \right ) \left ( 4 \sqrt{ 3 } - \left ( 3 \sqrt{ 5 } \right ) \right ) }$
 Expand using $\left(a + b\right)\left(c + d\right) = ac + ad + bc + bd$
$\dfrac { \left ( \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 3 } } \right ) \left ( \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 3 } } \right ) \color{#FF6800}{ + } \left ( \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 3 } } \right ) \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \left ( \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 5 } } \right ) \right ) \color{#FF6800}{ + } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 5 } } \right ) \left ( \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 3 } } \right ) \color{#FF6800}{ + } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 5 } } \right ) \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \left ( \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 5 } } \right ) \right ) } { \left ( 4 \sqrt{ 3 } + 3 \sqrt{ 5 } \right ) \left ( 4 \sqrt{ 3 } - \left ( 3 \sqrt{ 5 } \right ) \right ) }$
$\dfrac { \left ( 4 \sqrt{ 3 } \right ) \left ( 4 \sqrt{ 3 } \right ) + \left ( 4 \sqrt{ 3 } \right ) \times \left ( - \left ( 3 \sqrt{ 5 } \right ) \right ) + \left ( - 3 \sqrt{ 5 } \right ) \left ( 4 \sqrt{ 3 } \right ) + \left ( - 3 \sqrt{ 5 } \right ) \times \left ( - \left ( 3 \sqrt{ 5 } \right ) \right ) } { \left ( \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 5 } } \right ) \left ( \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 5 } } \right ) \right ) }$
 Expand the expression using $\left(a - b\right)\left(a + b\right) = a^{2} - b^{2}$
$\dfrac { \left ( 4 \sqrt{ 3 } \right ) \left ( 4 \sqrt{ 3 } \right ) + \left ( 4 \sqrt{ 3 } \right ) \times \left ( - \left ( 3 \sqrt{ 5 } \right ) \right ) + \left ( - 3 \sqrt{ 5 } \right ) \left ( 4 \sqrt{ 3 } \right ) + \left ( - 3 \sqrt{ 5 } \right ) \times \left ( - \left ( 3 \sqrt{ 5 } \right ) \right ) } { \left ( \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 3 } } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 2 } } }$
$\dfrac { \left ( 4 \sqrt{ 3 } \right ) \left ( 4 \sqrt{ 3 } \right ) + \left ( 4 \sqrt{ 3 } \right ) \times \left ( - \left ( 3 \sqrt{ 5 } \right ) \right ) + \left ( - 3 \sqrt{ 5 } \right ) \left ( 4 \sqrt{ 3 } \right ) + \left ( - 3 \sqrt{ 5 } \right ) \times \left ( - \left ( 3 \sqrt{ 5 } \right ) \right ) } { \left ( \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 3 } } \right ) ^ { \color{#FF6800}{ 2 } } - \left ( 3 \sqrt{ 5 } \right ) ^ { 2 } }$
 Calculate power 
$\dfrac { \left ( 4 \sqrt{ 3 } \right ) \left ( 4 \sqrt{ 3 } \right ) + \left ( 4 \sqrt{ 3 } \right ) \times \left ( - \left ( 3 \sqrt{ 5 } \right ) \right ) + \left ( - 3 \sqrt{ 5 } \right ) \left ( 4 \sqrt{ 3 } \right ) + \left ( - 3 \sqrt{ 5 } \right ) \times \left ( - \left ( 3 \sqrt{ 5 } \right ) \right ) } { \color{#FF6800}{ 48 } - \left ( 3 \sqrt{ 5 } \right ) ^ { 2 } }$
$\dfrac { \left ( 4 \sqrt{ 3 } \right ) \left ( 4 \sqrt{ 3 } \right ) + \left ( 4 \sqrt{ 3 } \right ) \times \left ( - \left ( 3 \sqrt{ 5 } \right ) \right ) + \left ( - 3 \sqrt{ 5 } \right ) \left ( 4 \sqrt{ 3 } \right ) + \left ( - 3 \sqrt{ 5 } \right ) \times \left ( - \left ( 3 \sqrt{ 5 } \right ) \right ) } { 48 - \left ( \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 2 } } }$
 Calculate power 
$\dfrac { \left ( 4 \sqrt{ 3 } \right ) \left ( 4 \sqrt{ 3 } \right ) + \left ( 4 \sqrt{ 3 } \right ) \times \left ( - \left ( 3 \sqrt{ 5 } \right ) \right ) + \left ( - 3 \sqrt{ 5 } \right ) \left ( 4 \sqrt{ 3 } \right ) + \left ( - 3 \sqrt{ 5 } \right ) \times \left ( - \left ( 3 \sqrt{ 5 } \right ) \right ) } { 48 - \color{#FF6800}{ 45 } }$
$\dfrac { \left ( \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 3 } } \right ) \left ( \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 3 } } \right ) + \left ( 4 \sqrt{ 3 } \right ) \times \left ( - \left ( 3 \sqrt{ 5 } \right ) \right ) + \left ( - 3 \sqrt{ 5 } \right ) \left ( 4 \sqrt{ 3 } \right ) + \left ( - 3 \sqrt{ 5 } \right ) \times \left ( - \left ( 3 \sqrt{ 5 } \right ) \right ) } { 48 - 45 }$
 Get rid of unnecessary parentheses 
$\dfrac { \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ \times } \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 3 } } + \left ( 4 \sqrt{ 3 } \right ) \times \left ( - \left ( 3 \sqrt{ 5 } \right ) \right ) + \left ( - 3 \sqrt{ 5 } \right ) \left ( 4 \sqrt{ 3 } \right ) + \left ( - 3 \sqrt{ 5 } \right ) \times \left ( - \left ( 3 \sqrt{ 5 } \right ) \right ) } { 48 - 45 }$
$\dfrac { \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ \times } \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 3 } } + \left ( 4 \sqrt{ 3 } \right ) \times \left ( - \left ( 3 \sqrt{ 5 } \right ) \right ) + \left ( - 3 \sqrt{ 5 } \right ) \left ( 4 \sqrt{ 3 } \right ) + \left ( - 3 \sqrt{ 5 } \right ) \times \left ( - \left ( 3 \sqrt{ 5 } \right ) \right ) } { 48 - 45 }$
 Simplify the expression 
$\dfrac { \color{#FF6800}{ 48 } + \left ( 4 \sqrt{ 3 } \right ) \times \left ( - \left ( 3 \sqrt{ 5 } \right ) \right ) + \left ( - 3 \sqrt{ 5 } \right ) \left ( 4 \sqrt{ 3 } \right ) + \left ( - 3 \sqrt{ 5 } \right ) \times \left ( - \left ( 3 \sqrt{ 5 } \right ) \right ) } { 48 - 45 }$
$\dfrac { 48 + \left ( \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 3 } } \right ) \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \left ( \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 5 } } \right ) \right ) + \left ( - 3 \sqrt{ 5 } \right ) \left ( 4 \sqrt{ 3 } \right ) + \left ( - 3 \sqrt{ 5 } \right ) \times \left ( - \left ( 3 \sqrt{ 5 } \right ) \right ) } { 48 - 45 }$
 Get rid of unnecessary parentheses 
$\dfrac { 48 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 5 } } + \left ( - 3 \sqrt{ 5 } \right ) \left ( 4 \sqrt{ 3 } \right ) + \left ( - 3 \sqrt{ 5 } \right ) \times \left ( - \left ( 3 \sqrt{ 5 } \right ) \right ) } { 48 - 45 }$
$\dfrac { 48 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 5 } } + \left ( - 3 \sqrt{ 5 } \right ) \left ( 4 \sqrt{ 3 } \right ) + \left ( - 3 \sqrt{ 5 } \right ) \times \left ( - \left ( 3 \sqrt{ 5 } \right ) \right ) } { 48 - 45 }$
 Simplify the expression 
$\dfrac { 48 \color{#FF6800}{ - } \color{#FF6800}{ 12 } \sqrt{ \color{#FF6800}{ 15 } } + \left ( - 3 \sqrt{ 5 } \right ) \left ( 4 \sqrt{ 3 } \right ) + \left ( - 3 \sqrt{ 5 } \right ) \times \left ( - \left ( 3 \sqrt{ 5 } \right ) \right ) } { 48 - 45 }$
$\dfrac { 48 - 12 \sqrt{ 15 } + \left ( \color{#FF6800}{ - } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 5 } } \right ) \left ( \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 3 } } \right ) + \left ( - 3 \sqrt{ 5 } \right ) \times \left ( - \left ( 3 \sqrt{ 5 } \right ) \right ) } { 48 - 45 }$
 Get rid of unnecessary parentheses 
$\dfrac { 48 - 12 \sqrt{ 15 } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ \times } \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 3 } } + \left ( - 3 \sqrt{ 5 } \right ) \times \left ( - \left ( 3 \sqrt{ 5 } \right ) \right ) } { 48 - 45 }$
$\dfrac { 48 - 12 \sqrt{ 15 } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ \times } \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 3 } } + \left ( - 3 \sqrt{ 5 } \right ) \times \left ( - \left ( 3 \sqrt{ 5 } \right ) \right ) } { 48 - 45 }$
 Simplify the expression 
$\dfrac { 48 - 12 \sqrt{ 15 } \color{#FF6800}{ - } \color{#FF6800}{ 12 } \sqrt{ \color{#FF6800}{ 15 } } + \left ( - 3 \sqrt{ 5 } \right ) \times \left ( - \left ( 3 \sqrt{ 5 } \right ) \right ) } { 48 - 45 }$
$\dfrac { 48 - 12 \sqrt{ 15 } - 12 \sqrt{ 15 } + \left ( \color{#FF6800}{ - } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 5 } } \right ) \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \left ( \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 5 } } \right ) \right ) } { 48 - 45 }$
 Get rid of unnecessary parentheses 
$\dfrac { 48 - 12 \sqrt{ 15 } - 12 \sqrt{ 15 } + \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 5 } } } { 48 - 45 }$
$\dfrac { 48 - 12 \sqrt{ 15 } - 12 \sqrt{ 15 } + \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 5 } } } { 48 - 45 }$
 Simplify the expression 
$\dfrac { 48 - 12 \sqrt{ 15 } - 12 \sqrt{ 15 } + \color{#FF6800}{ 45 } } { 48 - 45 }$
$\dfrac { 48 - 12 \sqrt{ 15 } - 12 \sqrt{ 15 } + 45 } { \color{#FF6800}{ 48 } \color{#FF6800}{ - } \color{#FF6800}{ 45 } }$
 Subtract $45$ from $48$
$\dfrac { 48 - 12 \sqrt{ 15 } - 12 \sqrt{ 15 } + 45 } { \color{#FF6800}{ 3 } }$
$\dfrac { \color{#FF6800}{ 48 } - 12 \sqrt{ 15 } - 12 \sqrt{ 15 } \color{#FF6800}{ + } \color{#FF6800}{ 45 } } { 3 }$
 Add $48$ and $45$
$\dfrac { \color{#FF6800}{ 93 } - 12 \sqrt{ 15 } - 12 \sqrt{ 15 } } { 3 }$
$\dfrac { 93 \color{#FF6800}{ - } \color{#FF6800}{ 12 } \sqrt{ \color{#FF6800}{ 15 } } \color{#FF6800}{ - } \color{#FF6800}{ 12 } \sqrt{ \color{#FF6800}{ 15 } } } { 3 }$
 Calculate between similar terms 
$\dfrac { 93 \color{#FF6800}{ - } \color{#FF6800}{ 24 } \sqrt{ \color{#FF6800}{ 15 } } } { 3 }$
$\color{#FF6800}{ \dfrac { 93 - 24 \sqrt{ 15 } } { 3 } }$
 Reduce the fraction 
$\color{#FF6800}{ 31 } \color{#FF6800}{ - } \color{#FF6800}{ 8 } \sqrt{ \color{#FF6800}{ 15 } }$
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