Is the square root of 18 rational and why?

Answer 1

No. √18 cannot be expressed as a fraction of the form p/q.

18 = 2 x 9 = 2 x 32

⇒ √18 = √(2 x 32)

= (√2) x 3.

So if √2 is rational then √18 is rational.

Assume √2 is rational.

Then p and q can be found such that √2 = p/q is in its simplest form, that is p and q have no common factor. Consider:

(√2)2 = (p/q)2

⇒ 2 = p2/q2

⇒ p2 = 2q2

Thus p2 is even, and so p must be even. Let p = 2r. Then:

p2 = (2r)2 = 2q2

⇒ 4r2 = 2q2

⇒ 2r2 = q2

Thus q2 is even, and so q must be even. Let q = 2s.

Thus p = 2r, q = 2s and so p and q have a common factor of 2.

But p and q are such that they have no common factor.

Contradiction.

Thus the assumption that √2 is rational is false, that is √2 is not rational, so √18 is not rational.