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Formula
Square root
Find the approximate value
$\sqrt{ 500000 }$
$500 \sqrt{ 2 }$
Find the square root
$\sqrt{ \color{#FF6800}{ 500000 } }$
 Do a factorization in prime factors for the integral number inside the radical sign 
$\sqrt{ \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 5 } } \color{#FF6800}{ \times } \color{#FF6800}{ 5 } ^ { \color{#FF6800}{ 6 } } }$
$\sqrt{ \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 5 } } \color{#FF6800}{ \times } \color{#FF6800}{ 5 } ^ { \color{#FF6800}{ 6 } } }$
 Separate the part that can be taken out from the radical sign and the part that cannot be taken out 
$\sqrt{ \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 4 } } \color{#FF6800}{ \times } \color{#FF6800}{ 5 } ^ { \color{#FF6800}{ 6 } } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 1 } } }$
$\sqrt{ \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 4 } } \times \color{#FF6800}{ 5 } ^ { \color{#FF6800}{ 6 } } \times \color{#FF6800}{ 2 } ^ { 1 } }$
 Separate the part that can be taken out of radical sign to different radical sign from the prime factor to the prime factor 
$\sqrt{ \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 4 } } } \sqrt{ \color{#FF6800}{ 5 } ^ { \color{#FF6800}{ 6 } } } \sqrt{ \color{#FF6800}{ 2 } ^ { 1 } }$
$\sqrt{ \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 4 } } } \sqrt{ 5 ^ { 6 } } \sqrt{ 2 ^ { 1 } }$
 Get rid of the radical sign and change it to the denominator of the exponent 
$\color{#FF6800}{ 2 } ^ { \color{#FF6800}{ \frac { 4 } { 2 } } } \sqrt{ 5 ^ { 6 } } \sqrt{ 2 ^ { 1 } }$
$2 ^ { \frac { 4 } { 2 } } \sqrt{ \color{#FF6800}{ 5 } ^ { \color{#FF6800}{ 6 } } } \sqrt{ 2 ^ { 1 } }$
 Get rid of the radical sign and change it to the denominator of the exponent 
$2 ^ { \frac { 4 } { 2 } } \times \color{#FF6800}{ 5 } ^ { \color{#FF6800}{ \frac { 6 } { 2 } } } \sqrt{ 2 ^ { 1 } }$
$2 ^ { \color{#FF6800}{ \frac { 4 } { 2 } } } \times 5 ^ { \frac { 6 } { 2 } } \sqrt{ 2 ^ { 1 } }$
 Reduce the fraction to the lowest term 
$2 ^ { \color{#FF6800}{ 2 } } \times 5 ^ { \frac { 6 } { 2 } } \sqrt{ 2 ^ { 1 } }$
$2 ^ { 2 } \times 5 ^ { \color{#FF6800}{ \frac { 6 } { 2 } } } \sqrt{ 2 ^ { 1 } }$
 Reduce the fraction to the lowest term 
$2 ^ { 2 } \times 5 ^ { \color{#FF6800}{ 3 } } \sqrt{ 2 ^ { 1 } }$
$2 ^ { 2 } \times 5 ^ { 3 } \sqrt{ 2 ^ { \color{#FF6800}{ 1 } } }$
 If the exponent is 1, get rid of it as it is unnecessary 
$2 ^ { 2 } \times 5 ^ { 3 } \sqrt{ 2 }$
$\color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \times 5 ^ { 3 } \sqrt{ 2 }$
 Calculate power 
$\color{#FF6800}{ 4 } \times 5 ^ { 3 } \sqrt{ 2 }$
$4 \times \color{#FF6800}{ 5 } ^ { \color{#FF6800}{ 3 } } \sqrt{ 2 }$
 Calculate power 
$4 \times \color{#FF6800}{ 125 } \sqrt{ 2 }$
$\color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 125 } \sqrt{ \color{#FF6800}{ 2 } }$
 Simplify the expression 
$\color{#FF6800}{ 500 } \sqrt{ \color{#FF6800}{ 2 } }$
$\approx 707.1068$
Find the approximate value
$\sqrt{ \color{#FF6800}{ 500000 } }$
 Find the approximate value of the square root 
$\approx \color{#FF6800}{ 707.1068 }$
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