Question

What's the square root of 48x^6?

Answer 1

Factor 48 into its prime factors
           48 = 24 • 3 

To simplify a square root, we extract factors which are squares, i.e., factors that are raised to an even exponent.

Factors which will be extracted are :
           16 = 24 

Factors which will remain inside the root are :
           3 = 3 

To complete this part of the simplification we take the squre root of the factors which are to be extracted. We do this by dividing their exponents by 2 :
           4 = 22 

At the end of this step the partly simplified SQRT looks like this: 
         4 • sqrt (3x3y4)  

Step  2  :Simplify the Variable part of the SQRT

Rules for simplifing variables which may be raised to a power:

   (1) variables with no exponent stay inside the radical
   (2) variables raised to power 1 or (-1) stay inside the radical
   (3) variables raised to an even exponent: Half the exponent taken out, nothing remains inside the radical. examples:
      (3.1) sqrt(x8)=x4
     (3.2) sqrt(x-6)=x-3

    (4) variables raised to an odd exponent which is  >2  or  <(-2) , examples:
      (4.1) sqrt(x5)=x2•sqrt(x)
     (4.2) sqrt(x-7)=x-3•sqrt(x-1)

 Applying these rules to our case we find out that 

      SQRT(x3y4) = xy2 • SQRT(x) 

Combine both simplifications

         sqrt (48x3y4) =
        4 xy2 • sqrt(3x) 

Simplified Root :      4 xy2 • sqrt(3x) 

Hope this helps~!
Happy studying~!
~{Dunsforhands}