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Show that the following numbers are irrational. `1/(sqrt(2))` (ii) `7sqrt(5)`

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`(1)/(sqrt(2)) = (1)/(sqrt(2)) xx (sqrt(2))/(sqrt(2)) = (sqrt(2))/(2) = (1)/(2) sqrt(2)`
Let, if possible `(1)/(sqrt(2))` is rational.
i.e., `(1)/(2) * sqrt(2)` is rational.
Now, `(1)/(2) * sqrt(2)` is rational and 2 is rational.
We know that the product of two rational numbers is rational
`therefore` `2 xx (1)/(2)sqrt(2)` is rational.
`rArr` `sqrt(2)` is rational.
This contradicts the fact that `sqrt(2)` is irrational.
So, our supposition is wrong.
Hence, `(1)/(sqrt(2))` is irrational. Hance Proved.
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