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

Question

NEED HELP ASAP!!

Answer 1

"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".

Answer 2

 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