Question

prove that √7 is an irrational​

Answer 1

Assume √7 is rational. So it can be expressed in the form p/q where p and q are integers & coprimes and q ≠ 0.

√7 = p/q

√7q = p

*squaring both sides*

7q² = p²

So p² is a multiple of 7.

So p is a multiple of 7.

So p = 7c

7q² = (7c)²

7q² = 49c²

q² = 7c²

So q² is a multiple of 7.

So q is a multiple of 7.

But we had said that p & q were coprimes.

So this assumption is false.

∴ √7 is irrational.