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Why is it important to find nonperfect roots in radical form to simplify the process of performing basic operations with radical expressions?

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2 Answers

"Why is it important to find nonperfect roots in radical form to simplify the process of performing basic operations with radical expressions?"
 
A "nonperfect" root must mean the root of a number that is not a perfect square.
 
Consider E = √(48) + √(175) - √(63). How would you simplify it?
 
First factorize each radicand into its prime factors:
 
E = √(2*2*2*3*3) + √(5*5*7) - √(3*3*7)
 
Every pair of identical primes under a radical represents a perfect square and can be replaced by one of those primes as a factor outside of the radical. Do that until there are no more pairs of primes under the radical and the radical that's left is a "nonperfect root".
 
E = 2*3*√(2) + 5*√(7) - 3*√(7)
 
And now you can identify and combine any "like terms":
 
E = 6√(2) + 2√(7)
 
So the short answer to the question is, "to identify and combine like terms".
 If I understand the question correctly.
 
  We can multiply and divide the radicals , but we can not add or subtract:
 
  ie,  √3 . √2 = √6.
 
     However:
 
     √3 + √2  , can not yield to another radical with a given radicand  such as √5
 
      Can only add radicals of equal radicand, ie.   √7 +2√7 = 3 √7.
 
      So , with this property of radicals, we can simplify the radicals, and if come up with multiple of simple radicals the we add.
 
     √50 + √18 - √8 =
 
     √(2 . 5 . 5 ) + √( 2 . 3 . 3 ) - √ (2 . 2 . 2) =
 
      √(2. 5^2)  +  √(2 .3^2)  -√ (  2 .2^2)  =
 
       √2 . √5^2  + √2 . √3^2  -  √2 . √2^2 =
 
        5 √2  + 3 √2 - 2√2 = 6 √2