Question 59835
The {{{sqrt(67)}}} would be rational if you could find an integer whose square is 67.
Let's see...

{{{8^2 = 64}}} Nope! Too small!
{{{9^2 = 81}}} Nope! Too large!

The answer is {{{sqrt(67) = 8.18535277187}}} (approx.) is irrational.
The second part of the problem is answered above.
Find two consecutive integers between which {{{sqrt(67)}}} lies on the number line.

{{{8 < sqrt(67) <9 }}}