# Show that there is no rational number whose square is $2$ or $8$ [duplicate]

Show that there is no rational number whose square is $$2$$ or $$8$$

Suppose there existes a rational $$\frac{a}{b}$$, with b nonzero, in reduced form, with a and b coprime such that its square was 2.
Then $$a^2=2b^2$$. Notice that if a number has any power (also 0) of 2 as a factor then its square has an even power of 2 as a factor. Thus, $$a^2$$ and $$b^2$$ have an even power of 2 as factors. But $$a^2=2b^2$$ and $$2b^2$$ has an odd power of 2 as factor. This is a contradiction for the prime decomposition of numbers is unique by the F.T.Arithmetic.