Solution - Simplifying square roots

Simplify :  sqrt(5v¹⁶) 

Step  1  :

Simplify the Integer part of the SQRT

Factor 5 into its prime factors
           5 = 5 

Note that 5 is a prime number, it only has itself as a factor (that is on top of the trivial factor "1")

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

In our case however, all the factors are only raised to the first power and this means that the square root can not be simplified

At the end of this step the partly simplified SQRT looks like this:
         sqrt (5v¹⁶)  

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(v¹⁶) = v⁸

Combine both simplifications

         sqrt (5v¹⁶) =
        v⁸ • sqrt(5) 

Simplified Root :