Question

prove that 3/2root5 is irrational

Answer 1

please give urgently

Answer 2

Suppose that \(\large \frac{3}{2\sqrt5}\) is rational. Then\[\dfrac{3}{2\sqrt5} = \dfrac{p}{q} \tag{p,q are coprime integers, q is not 0 }\]\[\dfrac{9}{10} = \dfrac{p^2}{q^2}\]\[9q^2 = 10p^2\]\[p^2 = \dfrac{9}{10} q^2\]So \(q^2\) is divisible by 10, meaning that \(q\) is a also divisible by 10. Let \(q = 10k\).\[p^2 = \dfrac{9}{10}100k^2\]\[p^2 = 90k^2 \]\[\dfrac{p^2}{10} = 9k^2\]So \(p^2\) is divisible by 10, meaning that \(p\) is also divisible by 10. Hence, p and q are both divisible by 10. But this is a contradiction to our original assumption that p,q are coprime. Hence proved.

Answer 3

@ParthKohli I think 2 root 5 whole squared is 4x5 = 20?