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How would I determine what's a radical expression?

How would I know if a radical expression can be simplified? Can I always represent a radical expression in exponential form? Can you give an example?

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Parviz F. | Mathematics professor at Community CollegesMathematics professor at Community Colle...
4.8 4.8 (4 lesson ratings) (4)
 Very simply radical expressions contain radical.
  Now we have to find out what are radicals, and what is their implication.
      Exponents are repeated multiplication:
        am = a. .a. .a.a,.  (m times) , like  2means  2 multiplied by intelf 4 times:
                                                           2 4 = 2 . 2 . 2. 2 = 16
           Radicals are the opposite of the exponents, and are designated by √----
               Now if we ask what number has multiplied by itself 4 times to result in 16, then we write as :
             4√16 = 2 , this read as 4th root of 16,
             2nd root is referred to as square root, it is the relationship between side of a square and the area
             3rd root is referred to as cube root, because corresponds the side of a cube and its volume.
              2nd root or square root, the number 2 is not written.
               √16 is read rad 16 or square root of 16
                 The general rule of definition of root is   n√X n = X , on this ground we can represent roots 
                    equivalent of fractional exponent.
                   n√X = X 1/n  
                    to find out the √16 = √(2. 2. 2. 2) = √24 =(24 ) 1/2 = 22 = 4
                     These are square roots of perfect square numbers, √17 = 4.12
                     √16 = 4   √17= 4.12          √25 =5 
Patricia S. | Math Tutoring for K-12 & CollegeMath Tutoring for K-12 & College
5.0 5.0 (39 lesson ratings) (39)
Those are good questions!
Q: How would I know if a radical expression can be simplified?
A radical expression can be simplified if the number or expression under the radical symbol has a factor that is a perfect root of whatever radical expression you're working with.  This might be easier to see in an example:
√(15x) --> The factors of 15x are 1, 3, 5, 15, x, 15x, 5x, and 3x.  However, none of those numbers are a perfect root of the radical we are working with (in this case, neither 5, 3, x, 5x, or 3x are perfect squares.)
√(24x) --> The factors of 16x are 2, 3, 4, 6, 8, 12, 24, x, etc. 4 is a perfect square, so we can write √(24x) as √(4)*√(6x), which can be simplified by taking the square root of 4 and rewriting the expression: 2*√(6x).
If your expression involves a third, fourth, fifth, etc. root, then the perfect root that you're looking for has to match the root in the expression.  For example, in a third root expression, 8 is a perfect third root (the third root of 8 is 2).
Q: Can I always represent a radical expression in exponential form?
As far as I know, yes.  The exponential form of any radical expression can be written as a fraction where the numerator is the exponent of the entire expression under the radical and the denominator is the root of the radical. Let's look at a couple examples:
√(x3) = x3/2
√(5x3) = (5x3)1/2 <--the exponent 3 only applies to the x, NOT the whole remaining expression under the radical (5x), so it is not the numerator of the fraction exponent.  Instead, the highest exponent that applies to both the 5 and the x is 1, so this is the numerator of the fraction exponent.
A radical expression that is taking the third root can be written as a fractional exponent where the denominator is 3.  The same type of this applies to fourth roots, fifth roots, etc.