SOLUTION: Find the square root of 32-6√15

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Question 1152514: Find the square root of 32-6√15
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


Consider this calculation:

%28sqrt%28a%29%2Bsqrt%28b%29%29%5E2

= %28a%2Bb%29%2B2sqrt%28ab%29

So if you have an expression in the form %28a%2Bb%29%2B2sqrt%28ab%29, then its square root is sqrt%28a%29%2Bsqrt%28b%29.

To find the square root of the given expression, you need to put it in exactly that form -- specifically, you need the radical part to be 2sqrt%28ab%29. So



And then the task is to find the numbers a and b for which a+b=32 and ab=135.

Those numbers are 27 and 5, so

sqrt%2832-6sqrt%2815%29%29+=+sqrt%2827%29-sqrt%285%29+=+3sqrt%283%29-sqrt%285%29

Answer by ikleyn(52835) About Me  (Show Source):
You can put this solution on YOUR website!
.

The key step is to notice (or to derive) that


    32 - 6%2Asqrt%2815%29 = %283%2Asqrt%283%29+-+sqrt%285%29%29%5E2.    (1)


Indeed, square the binomial  3%2Asqrt%283%29+-+sqrt%285%29,  using the formula   %28a-b%29%5E2 = a%5E2+-+2ab+%2B+b%5E2  with a = 3%2Asqrt%283%29  and  b = sqrt%285%29.


As soon as you established it, only one step remained to the answer:


    the square root of  32-6%2Asqrt%2815%29 is the square root of the right side of (1),  which (i.e. the square root)  is  3%2Asqrt%283%29+-+sqrt%285%29.    ANSWER