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Simplify sqrt(17 + 12*sqrt(2)).

 Nov 18, 2019
 #2
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sqrt [ 17 + 12sqrt (2)  ] 

 

Let us suppose that we can write this in the form a + b sqrt (2)    where a, b are both positive integers

 

 So

 

sqrt [ 17 + 12sqrt(2) ]  = a + b sqrt (2)       square both sides

 

17 + 12 sqrt (2)  =  a^2 + 2ab sqrt (2)  + 2b^2

 

Equating coefficients it must be that

 

12sqrt (2)  =  2ab sqrt (2)         and     a^2 + 2b^2  = 17      (2)

12  = 2ab

6  = ab

b = 6/a      (1)

 

 

Sub (1)  into (2)  and we have that

 

a^2 + 2 (6/a)^2 =17

 

a^2 + 2*36 / a^2  = 17

 

a^2 + 72/a^2  = 17       multiply through by a^2

 

a^4 + 72  = 17a^2

 

a^4 - 17a^2 + 72  =  0        factor

 

(a^2 - 8) ( a^2 - 9)  =  0

 

We want  a  to be a positive integer  so  a =  3

 

And b = 6/a=  6/3  =2

 

So.....sqrt [ 17 + 12sqrt (2)  ]  can be simplified to

 

3 + 2sqrt (2)

 

 

 

cool cool cool

 Nov 18, 2019

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