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 \(\sqrt{53+20\sqrt{7}}\) can be written in the form \(a+b\sqrt{c}\), where \(a, b\)  and \(c\) are integers and \(c\) has no factors which is a perfect square of any positive integer other than 1. Find \(a+b+c\).

 Jan 16, 2020
 #1
avatar+107405 
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√ [53 + 20√ 7]  =   a + b√ c      square both sides

 

53 + 20√ 7  =  a^2 +2ab√ c  + b^2c

 

Equating terms

 

2ab =  20          a^2  +  b^2c   =  53

ab  =10             a^2 + 7b^2  =53

And c  = 7             

 

Then

a^2 + 7b^2  = 53

Let b =  2

a^2 + 7(2)^2  = 53

a^2 + 28 = 53

a^2  = 25

 a  = 5

 

So

 

a + b√ c  =   5 + 2√ 7

 

So

 

a + b + c  =   5 +  2 + 7   =  14

 

cool cool cool

 Jan 16, 2020

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