Question

show 2-3root5 is an irrational

Answer 1

Answer

proof that root 5 is irrational: 

assume that root 5 is rational 

root 5 = p/q 
where p and q are integers and p/q is in lowest terms. 
5 = p^2/q^2 
5 q^2 = p^2 

5 divides p^2 
if 5 divides p^2 then since 5 is prime, 5 must divide p 
r = p/5 
q^2 = 5 r^2 

Which means the 5 must divide q, but if 5 divides p and 5 divides q then p/q is not in lowest terms. 
we have a contradiction 
sqrt 5 does not equal p/q 
sqrt 5 is irrational 

proof that 2 - 3 root 5 is irrational 

assume that 
2 - 3 root 5 is rational 
2 - 3 root 5 = r (where r is a rational number 
root 5 = (2-r)/3 
if r is rational then (2-r)/3 is rational 

but root 5 is irrational. 

2 - 3 root 5 is irr