SOLUTION: If z = 2<sup>14</sup>&#8729;3<sup>20</sup>, what is the largest power of 12 that evenly divides into the square root of z?

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Question 1049891: If z = 214∙320, what is the largest
power of 12 that evenly divides into the square root of z?
Found 2 solutions by Alan3354, Edwin McCravy:
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
z = 2^14*3^20

--> 2^7
==================
Captain Obvious is correct, my answer is not right.
I misread it, thought it said power of 2.
---
I wouldn't try to make a Master's thesis out of it, tho.
Answer by Edwin McCravy(20060)   (Show Source): You can put this solution on YOUR website!
The solution above is incorrect!
If z = 214∙320, what is the largest 
power of 12 that evenly divides into the square root of z?




Let p = the largest power of 12 that evenly divides into the
square root of z





We are given:



We find the square root of z





Now we divide  INTO , which amounts to
dividing  BY 



Subtracting exponents on the right



Negative exponents cause denominators. Therefore both those 
exponents must be non-negative (greater than or equal to zero) 
in order for the above result to be an integer, so we have:

 and 

 and 

Remember that the inequality reverses when dividing 
by a negative number:

 and 

  and 

 and 

The largest integer value of p that satisfies both those is p = 3.

Answer: The largest power of 12 that evenly divides into the square 
root of z is 3.

Edwin




 

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