Solution - Simplifying square roots

Simplify :  sqrt(324x⁶y¹⁰) 

Step  1  :

Simplify the Integer part of the SQRT

Factor 324 into its prime factors
           324 = 2² • 3⁴ 

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 :
           324 = 2² • 3⁴ 

No factors remain inside the root !!

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 :
           18 = 2 • 3² 

At the end of this step the partly simplified SQRT looks like this:
         18 sqrt(x⁶y¹⁰)

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(x⁶y¹⁰) = x³y⁵

Combine both simplifications

         sqrt (324x⁶y¹⁰) =
        18 x³y⁵ 

Simplified Root :