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$\sqrt{ \color{#FF6800}{ 507 } }$

$ $ Do a factorization in prime factors for the integral number inside the radical sign $ $

$\sqrt{ \color{#FF6800}{ 3 } \color{#FF6800}{ \times } \color{#FF6800}{ 13 } ^ { \color{#FF6800}{ 2 } } }$

$\sqrt{ \color{#FF6800}{ 3 } \color{#FF6800}{ \times } \color{#FF6800}{ 13 } ^ { \color{#FF6800}{ 2 } } }$

$ $ Separate the part that can be taken out from the radical sign and the part that cannot be taken out $ $

$\sqrt{ \color{#FF6800}{ 13 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } }$

$\sqrt{ \color{#FF6800}{ 13 } ^ { \color{#FF6800}{ 2 } } \times 3 }$

$ $ Separate the part that can be taken out of radical sign to different radical sign from the prime factor to the prime factor $ $

$\sqrt{ \color{#FF6800}{ 13 } ^ { \color{#FF6800}{ 2 } } } \sqrt{ 3 }$

$\sqrt{ \color{#FF6800}{ 13 } ^ { \color{#FF6800}{ 2 } } } \sqrt{ 3 }$

$ $ Get rid of the radical sign and change it to the denominator of the exponent $ $

$\color{#FF6800}{ 13 } ^ { \color{#FF6800}{ \frac { 2 } { 2 } } } \sqrt{ 3 }$

$13 ^ { \color{#FF6800}{ \frac { 2 } { 2 } } } \sqrt{ 3 }$

$ $ Reduce the fraction to the lowest term $ $

$13 ^ { \color{#FF6800}{ 1 } } \sqrt{ 3 }$

$13 ^ { \color{#FF6800}{ 1 } } \sqrt{ 3 }$

$ $ If the exponent is 1, get rid of it as it is unnecessary $ $

$13 \sqrt{ 3 }$

$\sqrt{ \color{#FF6800}{ 507 } }$

$ $ Find the approximate value of the square root $ $

$\approx \color{#FF6800}{ 22.5167 }$