Question

Show that `2sqrt(7)` is irrational.

Answer 1

`2/sqrt(7) = 2/sqrt(7) xx sqrt(7)/sqrt(7) = 2/7 sqrt(7)`

Let `2/7 sqrt (7)` is a rational number.

∴ `2/7 sqrt (7) = p/q, `where p and q are some integers and HCF(p,q) = 1     ….(1)

⇒2`sqrt(7)`q = 7p

⇒(2`sqrt(7) q) ^ 2 = (7p)^ 2`
⇒`7(4q^2) = 49p^2`
⇒`4q^2 = 7p^2`
⇒ `q^2 `is divisible by 7
⇒ q is divisible by 7        …..(2)
Let q = 7m, where m is some integer.
∴2`sqrt(7)` q = 7p
⇒ [2`sqrt(7) (7m)]^2 = (7p)^2`
⇒`343(4m^2) = 49p^2`
⇒` 7(4m^2) = p^2`
⇒ `p^2` is divisible by 7
⇒ p is divisible by 7             ….(3)
From (2) and (3), 7 is a common factor of both p and q, which contradicts (1).
Hence, our assumption is wrong.
Thus, `2 sqrt(7)` is irrational.