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Question

if A=5+  2 root 6 find the value of rootA + 1/root A


Solution

 A = 5 + 2√6 
      
       = 3 + 2(√3)×(√2) + 2   ←←← Since √(ab)=(√a)(√b) 
         
        = (√3)² + 2×(√3)×(√2) × (√2)²   ←←← Since a=(√a)² 
       
        = (√3 + √2)²    ←←← Since x²+2xy+y²=(x+y)² 

Thus: 
  
 √A = √3 + √2 

And: 
   1/√A = 1/(√3 + √2) 
      = (√3 – √2) / [(√3 + √2)(√3 – √2)]   ←←← Since (1/x)=y/(xy) 
      = (√3 – √2) / [(√3)² – (√2)²]   ←←← Since (x+y)(x–y)=x²–y² 
      = (√3 – √2) / (3 – 2)    ←←← Since (√a)²=a 
      = (√3 – √2) / 1 
      = √3 – √2 

Thus finally: 
   √A + 1/√A = (√3 + √2) + (√3 – √2) = 2√3 

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