If you square a number and reduce the square to it's prime factors those prime factors will have even exponents only if it's a rational number, therefore if you multiply a number reduce the square to it's prime factors and those prime factors have odd exponents then the number is irrational.

Here is that process with a rational number:

7/10 (which we all know is exactly 0.7 and therefore rational)

Square the number
= (7/10)²
Reduce numbers to prime factors
=7²/(5x2)²
Expand
=7²/(5²x2²)

All components are prime, all exponents are even therefore it is a rational number.

Let's try with root 2:

To prove that root 2 is irrational we'll use the same system.

We have to twist it around a bit but this does make it easier, If root 2 was rational then we could represent it by the fraction a/b a ratio reduced to its lowest terms.

(root 2) = a/b

Our first step is to square that number
(root 2)²=(a/b)²
Simplified
2=a²/b²
Multiply both sides by b²
2b²=a²

This shows a² is even therefore a must be even - as we all know of old the only numbers that are even when squared are even numbers.

a can be represented by the expression 2c (c being half a). Let's replace a in the formula with 2c... we get

2b² = (2c)²

Or

2b² = 4c²

b² = 2c²

So b must be even too.

We have now proved that both a and b are multiples of 2, but this can't be as the premise is that the expression a/b is reduced to its lowest terms and we have shown that both terms have a factor of 2.

This shows that root 2 cannot be represented by the fraction a/b and so proves that it is irrational.