Solution - Simplifying square roots

Simplify :  sqrt(162x⁵) 

Step  1  :

Simplify the Integer part of the SQRT

Factor 162 into its prime factors
           162 = 2 • 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 :
           81 = 3⁴ 

Factors which will remain inside the root are :
           2 = 2 

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 :
           9 = 3² 

At the end of this step the partly simplified SQRT looks like this:
         9 • sqrt (2x⁵)  

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(x⁵) = x² • SQRT(x) 

Combine both simplifications

         sqrt (162x⁵) =
        9 x² • sqrt(2x) 

Simplified Root :