$\sqrt{ \color{#FF6800}{ 147 } }$
$ $ Do a factorization in prime factors for the integral number inside the radical sign $ $
$\sqrt{ \color{#FF6800}{ 3 } \color{#FF6800}{ \times } \color{#FF6800}{ 7 } ^ { \color{#FF6800}{ 2 } } }$
$\sqrt{ \color{#FF6800}{ 3 } \color{#FF6800}{ \times } \color{#FF6800}{ 7 } ^ { \color{#FF6800}{ 2 } } }$
$ $ Separate the part that can be taken out from the radical sign and the part that cannot be taken out $ $
$\sqrt{ \color{#FF6800}{ 7 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } }$
$\sqrt{ \color{#FF6800}{ 7 } ^ { \color{#FF6800}{ 2 } } \times 3 }$
$ $ Separate the part that can be taken out of radical sign to different radical sign from the prime factor to the prime factor $ $
$\sqrt{ \color{#FF6800}{ 7 } ^ { \color{#FF6800}{ 2 } } } \sqrt{ 3 }$
$\sqrt{ \color{#FF6800}{ 7 } ^ { \color{#FF6800}{ 2 } } } \sqrt{ 3 }$
$ $ Get rid of the radical sign and change it to the denominator of the exponent $ $
$\color{#FF6800}{ 7 } ^ { \color{#FF6800}{ \frac { \color{#FF6800}{ 2 } } { \color{#FF6800}{ 2 } } } } \sqrt{ 3 }$
$7 ^ { \color{#FF6800}{ \frac { \color{#FF6800}{ 2 } } { \color{#FF6800}{ 2 } } } } \sqrt{ 3 }$
$ $ Reduce the fraction to the lowest term $ $
$7 ^ { \color{#FF6800}{ 1 } } \sqrt{ 3 }$
$7 ^ { \color{#FF6800}{ 1 } } \sqrt{ 3 }$
$ $ If the exponent is 1, get rid of it as it is unnecessary $ $
$7 \sqrt{ 3 }$