InterviewSolution
Saved Bookmarks
InterviewSolution
| 1. |
Prove that ` (sqrt2+ 5sqrt2)` is irrational. |
|
Answer» Let us assume that ` ( sqrt2 +sqrt3)` is irrational. Then, there exist co-primes a and b such that ` sqrt2 + sqrt3= a/b` ` Rightarrow sqrt3 = a/b -sqrt2` ` Rightarrow (sqrt3)^(2) = ( a/b -sqrt2)^(2)` ` Rightarrow 3 = a^(2)/b^(2) - (2a)/b sqrt2 +2` ` Rightarrow (2a)/b sqrt2 = a^(2)/b^(2) -1` ` Rightarrow sqrt2 = (a^(2) -b^(2))/(2ab)` Since a and b are intergers, so ` (a^(2) -b^(2))/(2ab)` is rational. Thus, ` sqrt2` is also rational. But, this contradicts the fact that ` sqrt2` is irrational , so, our assumption is incorrect. Hence ` (sqrt2+ sqrt3)` is irrational. |
|