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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?"

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".

Parviz F. | Mathematics professor at Community CollegesMathematics professor at Community Colle...

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

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