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Simplifying square roots

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sqrt(24a^9b^14)

This solution deals with simplifying square roots.

Solution found

2a^4b^7*sqrt(6a)
2a^4b^7*sqrt(6a)

Step by Step Solution

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Simplify :  sqrt(24a9b14

Step  1  :

Simplify the Integer part of the SQRT

Factor 24 into its prime factors
           24 = 23 • 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 :
           4 = 22 

Factors which will remain inside the root are :
           6 = 2 • 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 :
           2 = 2 

At the end of this step the partly simplified SQRT looks like this:
         2 • sqrt (6a9b14)  

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(a9b14) = a4b7 • SQRT(a) 

Combine both simplifications

         sqrt (24a9b14) =
        2 a4b7 • sqrt(6a) 


Simplified Root :

      2 a4b7 • sqrt(6a) 

Why learn this

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