Question

If A = 5+2√6, then find the value of √A + 1/√A

Answer 1

You have

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
hope this helps