# Calculator search results

Formula
Calculate the value
$\dfrac{ 2- \sqrt{ 5 } }{ 2+ \sqrt{ 5 } } + \dfrac{ 2+ \sqrt{ 5 } }{ 2- \sqrt{ 5 } }$
$- 18$
Calculate the value
$\dfrac { 2 - \sqrt{ 5 } } { 2 + \sqrt{ 5 } } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
 Find the conjugate irrational number of denominator 
$\color{#FF6800}{ \dfrac { 2 - \sqrt{ 5 } } { 2 + \sqrt{ 5 } } } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 2 - \sqrt{ 5 } } { 2 - \sqrt{ 5 } } } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$\dfrac { 2 - \sqrt{ 5 } } { 2 + \sqrt{ 5 } } \times \dfrac { 2 - \sqrt{ 5 } } { 2 - \sqrt{ 5 } } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
 The denominator is multiplied by denominator, and the numerator is multiplied by numerator 
$\color{#FF6800}{ \dfrac { \left ( 2 - \sqrt{ 5 } \right ) \left ( 2 - \sqrt{ 5 } \right ) } { \left ( 2 + \sqrt{ 5 } \right ) \left ( 2 - \sqrt{ 5 } \right ) } } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$\dfrac { \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } \right ) \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } \right ) } { \left ( 2 + \sqrt{ 5 } \right ) \left ( 2 - \sqrt{ 5 } \right ) } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
 Expand using $\left(a + b\right)\left(c + d\right) = ac + ad + bc + bd$
$\dfrac { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } \right ) \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } \right ) } { \left ( 2 + \sqrt{ 5 } \right ) \left ( 2 - \sqrt{ 5 } \right ) } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$\dfrac { 2 \times 2 + 2 \times \left ( - \sqrt{ 5 } \right ) - \sqrt{ 5 } \times 2 - \sqrt{ 5 } \times \left ( - \sqrt{ 5 } \right ) } { \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 5 } } \right ) \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } \right ) } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
 Expand the expression using $\left(a - b\right)\left(a + b\right) = a^{2} - b^{2}$
$\dfrac { 2 \times 2 + 2 \times \left ( - \sqrt{ 5 } \right ) - \sqrt{ 5 } \times 2 - \sqrt{ 5 } \times \left ( - \sqrt{ 5 } \right ) } { \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \left ( \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 2 } } } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$\dfrac { 2 \times 2 + 2 \times \left ( - \sqrt{ 5 } \right ) - \sqrt{ 5 } \times 2 - \sqrt{ 5 } \times \left ( - \sqrt{ 5 } \right ) } { \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } - \left ( \sqrt{ 5 } \right ) ^ { 2 } } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
 Calculate power 
$\dfrac { 2 \times 2 + 2 \times \left ( - \sqrt{ 5 } \right ) - \sqrt{ 5 } \times 2 - \sqrt{ 5 } \times \left ( - \sqrt{ 5 } \right ) } { \color{#FF6800}{ 4 } - \left ( \sqrt{ 5 } \right ) ^ { 2 } } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$\dfrac { 2 \times 2 + 2 \times \left ( - \sqrt{ 5 } \right ) - \sqrt{ 5 } \times 2 - \sqrt{ 5 } \times \left ( - \sqrt{ 5 } \right ) } { 4 - \left ( \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 2 } } } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
 Calculate power 
$\dfrac { 2 \times 2 + 2 \times \left ( - \sqrt{ 5 } \right ) - \sqrt{ 5 } \times 2 - \sqrt{ 5 } \times \left ( - \sqrt{ 5 } \right ) } { 4 - \color{#FF6800}{ 5 } } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$\dfrac { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } + 2 \times \left ( - \sqrt{ 5 } \right ) - \sqrt{ 5 } \times 2 - \sqrt{ 5 } \times \left ( - \sqrt{ 5 } \right ) } { 4 - 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
 Multiply $2$ and $2$
$\dfrac { \color{#FF6800}{ 4 } + 2 \times \left ( - \sqrt{ 5 } \right ) - \sqrt{ 5 } \times 2 - \sqrt{ 5 } \times \left ( - \sqrt{ 5 } \right ) } { 4 - 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$\dfrac { 4 + \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } \right ) - \sqrt{ 5 } \times 2 - \sqrt{ 5 } \times \left ( - \sqrt{ 5 } \right ) } { 4 - 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
 Simplify the expression 
$\dfrac { 4 \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 5 } } - \sqrt{ 5 } \times 2 - \sqrt{ 5 } \times \left ( - \sqrt{ 5 } \right ) } { 4 - 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$\dfrac { 4 - 2 \sqrt{ 5 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } - \sqrt{ 5 } \times \left ( - \sqrt{ 5 } \right ) } { 4 - 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
 Simplify the expression 
$\dfrac { 4 - 2 \sqrt{ 5 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 5 } } - \sqrt{ 5 } \times \left ( - \sqrt{ 5 } \right ) } { 4 - 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$\dfrac { 4 - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } \color{#FF6800}{ - } \sqrt{ 5 } \times \left ( \color{#FF6800}{ - } \sqrt{ 5 } \right ) } { 4 - 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
 Since negative numbers are multiplied by an even number, remove the (-) sign 
$\dfrac { 4 - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } + \sqrt{ 5 } \sqrt{ 5 } } { 4 - 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$\dfrac { 4 - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } + \sqrt{ \color{#FF6800}{ 5 } } \sqrt{ 5 } } { 4 - 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
 If the exponent is omitted, the exponent of that term is equal to 1 
$\dfrac { 4 - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } + \left ( \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 1 } } \sqrt{ 5 } } { 4 - 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$\dfrac { 4 - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } + \left ( \sqrt{ 5 } \right ) ^ { 1 } \sqrt{ \color{#FF6800}{ 5 } } } { 4 - 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
 If the exponent is omitted, the exponent of that term is equal to 1 
$\dfrac { 4 - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } + \left ( \sqrt{ 5 } \right ) ^ { 1 } \left ( \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 1 } } } { 4 - 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$\dfrac { 4 - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } + \left ( \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 1 } } \left ( \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 1 } } } { 4 - 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
 Add the exponent as the base is the same 
$\dfrac { 4 - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } + \left ( \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } } } { 4 - 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$\dfrac { 4 - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } + \left ( \sqrt{ 5 } \right ) ^ { \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } } } { 4 - 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
 Add $1$ and $1$
$\dfrac { 4 - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } + \left ( \sqrt{ 5 } \right ) ^ { \color{#FF6800}{ 2 } } } { 4 - 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$\dfrac { 4 - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } + \left ( \sqrt{ 5 } \right ) ^ { 2 } } { \color{#FF6800}{ 4 } \color{#FF6800}{ - } \color{#FF6800}{ 5 } } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
 Subtract $5$ from $4$
$\dfrac { 4 - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } + \left ( \sqrt{ 5 } \right ) ^ { 2 } } { \color{#FF6800}{ - } \color{#FF6800}{ 1 } } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$\dfrac { 4 - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } + \left ( \sqrt{ 5 } \right ) ^ { 2 } } { \color{#FF6800}{ - } 1 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
 If the denominator is 1, the denominator can be removed 
$\color{#FF6800}{ - } \left ( \color{#FF6800}{ 4 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ + } \left ( \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 2 } } \right ) + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$- \left ( 4 - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } + \left ( \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 2 } } \right ) + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
 If you square the radical sign, it will disappear 
$- \left ( 4 - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } + \color{#FF6800}{ 5 } \right ) + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$- \left ( \color{#FF6800}{ 4 } - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } \color{#FF6800}{ + } \color{#FF6800}{ 5 } \right ) + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
 Add $4$ and $5$
$- \left ( \color{#FF6800}{ 9 } - 2 \sqrt{ 5 } - 2 \sqrt{ 5 } \right ) + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$- \left ( 9 \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 5 } } \right ) + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
 Calculate between similar terms 
$- \left ( 9 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 5 } } \right ) + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$\color{#FF6800}{ - } \left ( \color{#FF6800}{ 9 } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 5 } } \right ) + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
 Change the symbol of each term in parentheses when there is a (-) symbol in front of parentheses 
$\color{#FF6800}{ - } \color{#FF6800}{ 9 } + \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 5 } } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
$- 9 + 4 \sqrt{ 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } }$
 Find the conjugate irrational number of denominator 
$- 9 + 4 \sqrt{ 5 } + \color{#FF6800}{ \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } } } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 2 + \sqrt{ 5 } } { 2 + \sqrt{ 5 } } }$
$- 9 + 4 \sqrt{ 5 } + \dfrac { 2 + \sqrt{ 5 } } { 2 - \sqrt{ 5 } } \times \dfrac { 2 + \sqrt{ 5 } } { 2 + \sqrt{ 5 } }$
 The denominator is multiplied by denominator, and the numerator is multiplied by numerator 
$- 9 + 4 \sqrt{ 5 } + \color{#FF6800}{ \dfrac { \left ( 2 + \sqrt{ 5 } \right ) \left ( 2 + \sqrt{ 5 } \right ) } { \left ( 2 - \sqrt{ 5 } \right ) \left ( 2 + \sqrt{ 5 } \right ) } }$
$- 9 + 4 \sqrt{ 5 } + \dfrac { \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 5 } } \right ) \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 5 } } \right ) } { \left ( 2 - \sqrt{ 5 } \right ) \left ( 2 + \sqrt{ 5 } \right ) }$
 Expand using $\left(a + b\right)\left(c + d\right) = ac + ad + bc + bd$
$- 9 + 4 \sqrt{ 5 } + \dfrac { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 5 } } \sqrt{ \color{#FF6800}{ 5 } } } { \left ( 2 - \sqrt{ 5 } \right ) \left ( 2 + \sqrt{ 5 } \right ) }$
$- 9 + 4 \sqrt{ 5 } + \dfrac { 2 \times 2 + 2 \sqrt{ 5 } + \sqrt{ 5 } \times 2 + \sqrt{ 5 } \sqrt{ 5 } } { \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } \right ) \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 5 } } \right ) }$
 Expand the expression using $\left(a - b\right)\left(a + b\right) = a^{2} - b^{2}$
$- 9 + 4 \sqrt{ 5 } + \dfrac { 2 \times 2 + 2 \sqrt{ 5 } + \sqrt{ 5 } \times 2 + \sqrt{ 5 } \sqrt{ 5 } } { \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \left ( \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 2 } } }$
$- 9 + 4 \sqrt{ 5 } + \dfrac { 2 \times 2 + 2 \sqrt{ 5 } + \sqrt{ 5 } \times 2 + \sqrt{ \color{#FF6800}{ 5 } } \sqrt{ \color{#FF6800}{ 5 } } } { 2 ^ { 2 } - \left ( \sqrt{ 5 } \right ) ^ { 2 } }$
 Arrange the expression 
$- 9 + 4 \sqrt{ 5 } + \dfrac { 2 \times 2 + 2 \sqrt{ 5 } + \sqrt{ 5 } \times 2 + \sqrt{ \color{#FF6800}{ 5 } \color{#FF6800}{ \times } \color{#FF6800}{ 5 } } } { 2 ^ { 2 } - \left ( \sqrt{ 5 } \right ) ^ { 2 } }$
$- 9 + 4 \sqrt{ 5 } + \dfrac { 2 \times 2 + 2 \sqrt{ 5 } + \sqrt{ 5 } \times 2 + \sqrt{ 5 \times 5 } } { \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } - \left ( \sqrt{ 5 } \right ) ^ { 2 } }$
 Calculate power 
$- 9 + 4 \sqrt{ 5 } + \dfrac { 2 \times 2 + 2 \sqrt{ 5 } + \sqrt{ 5 } \times 2 + \sqrt{ 5 \times 5 } } { \color{#FF6800}{ 4 } - \left ( \sqrt{ 5 } \right ) ^ { 2 } }$
$- 9 + 4 \sqrt{ 5 } + \dfrac { 2 \times 2 + 2 \sqrt{ 5 } + \sqrt{ 5 } \times 2 + \sqrt{ 5 \times 5 } } { 4 - \left ( \sqrt{ \color{#FF6800}{ 5 } } \right ) ^ { \color{#FF6800}{ 2 } } }$
 Calculate power 
$- 9 + 4 \sqrt{ 5 } + \dfrac { 2 \times 2 + 2 \sqrt{ 5 } + \sqrt{ 5 } \times 2 + \sqrt{ 5 \times 5 } } { 4 - \color{#FF6800}{ 5 } }$
$- 9 + 4 \sqrt{ 5 } + \dfrac { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } + 2 \sqrt{ 5 } + \sqrt{ 5 } \times 2 + \sqrt{ 5 \times 5 } } { 4 - 5 }$
 Multiply $2$ and $2$
$- 9 + 4 \sqrt{ 5 } + \dfrac { \color{#FF6800}{ 4 } + 2 \sqrt{ 5 } + \sqrt{ 5 } \times 2 + \sqrt{ 5 \times 5 } } { 4 - 5 }$
$- 9 + 4 \sqrt{ 5 } + \dfrac { 4 + 2 \sqrt{ 5 } + \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } + \sqrt{ 5 \times 5 } } { 4 - 5 }$
 Simplify the expression 
$- 9 + 4 \sqrt{ 5 } + \dfrac { 4 + 2 \sqrt{ 5 } + \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 5 } } + \sqrt{ 5 \times 5 } } { 4 - 5 }$
$- 9 + 4 \sqrt{ 5 } + \dfrac { 4 + 2 \sqrt{ 5 } + 2 \sqrt{ 5 } + \sqrt{ \color{#FF6800}{ 5 } \color{#FF6800}{ \times } \color{#FF6800}{ 5 } } } { 4 - 5 }$
 Multiply $5$ and $5$
$- 9 + 4 \sqrt{ 5 } + \dfrac { 4 + 2 \sqrt{ 5 } + 2 \sqrt{ 5 } + \sqrt{ \color{#FF6800}{ 25 } } } { 4 - 5 }$
$- 9 + 4 \sqrt{ 5 } + \dfrac { 4 + 2 \sqrt{ 5 } + 2 \sqrt{ 5 } + \sqrt{ 25 } } { \color{#FF6800}{ 4 } \color{#FF6800}{ - } \color{#FF6800}{ 5 } }$
 Subtract $5$ from $4$
$- 9 + 4 \sqrt{ 5 } + \dfrac { 4 + 2 \sqrt{ 5 } + 2 \sqrt{ 5 } + \sqrt{ 25 } } { \color{#FF6800}{ - } \color{#FF6800}{ 1 } }$
$- 9 + 4 \sqrt{ 5 } + \dfrac { 4 + 2 \sqrt{ 5 } + 2 \sqrt{ 5 } + \sqrt{ 25 } } { \color{#FF6800}{ - } 1 }$
 If the denominator is 1, the denominator can be removed 
$- 9 + 4 \sqrt{ 5 } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 4 } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 25 } } \right )$
$- 9 + 4 \sqrt{ 5 } - \left ( 4 + 2 \sqrt{ 5 } + 2 \sqrt{ 5 } + \sqrt{ \color{#FF6800}{ 25 } } \right )$
 Organize the part that can be taken out of the radical sign inside the square root symbol 
$- 9 + 4 \sqrt{ 5 } - \left ( 4 + 2 \sqrt{ 5 } + 2 \sqrt{ 5 } + \color{#FF6800}{ 5 } \right )$
$- 9 + 4 \sqrt{ 5 } - \left ( \color{#FF6800}{ 4 } + 2 \sqrt{ 5 } + 2 \sqrt{ 5 } \color{#FF6800}{ + } \color{#FF6800}{ 5 } \right )$
 Add $4$ and $5$
$- 9 + 4 \sqrt{ 5 } - \left ( \color{#FF6800}{ 9 } + 2 \sqrt{ 5 } + 2 \sqrt{ 5 } \right )$
$- 9 + 4 \sqrt{ 5 } - \left ( 9 + \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 5 } } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 5 } } \right )$
 Calculate between similar terms 
$- 9 + 4 \sqrt{ 5 } - \left ( 9 + \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 5 } } \right )$
$- 9 + 4 \sqrt{ 5 } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 9 } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 5 } } \right )$
 Change the symbol of each term in parentheses when there is a (-) symbol in front of parentheses 
$- 9 + 4 \sqrt{ 5 } \color{#FF6800}{ - } \color{#FF6800}{ 9 } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 5 } }$
$- 9 \color{#FF6800}{ + } \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 5 } } - 9 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 5 } }$
 Eliminate opponent number 
$- 9 - 9$
$\color{#FF6800}{ - } \color{#FF6800}{ 9 } \color{#FF6800}{ - } \color{#FF6800}{ 9 }$
 Find the sum of the negative numbers 
$\color{#FF6800}{ - } \color{#FF6800}{ 18 }$
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