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Simplifying square roots

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This solution deals with simplifying square roots.

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Step by Step Solution

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Simplify :  sqrt(18a3

Step  1  :

Simplify the Integer part of the SQRT

Factor 18 into its prime factors
           18 = 2 • 32 

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 :
           9 = 32 

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 :
           3 = 3 

At the end of this step the partly simplified SQRT looks like this:
         3 • sqrt (2a3)  

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(a3) = a • SQRT(a) 

Combine both simplifications

         sqrt (18a3) =
        3 a • sqrt(2a) 

Simplified Root :

      3 a • sqrt(2a) 

Why learn this

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