Question

prove that 4 minus 5 root 2 is an irrational number

Answer 1

Consider, 4−3√2
Let 4−3√2 = (a/b) a rational number 
⇒ −3√2 = (a/b) − 4
⇒ −3√2 = (a − 4b)/b
⇒ √2 = (a − 4b)/(−3b)
Since a, b are integers, then (a − 4b)/(−3b) represents a rational number.
But this is a contradiction since RHS is a rational number where as LHS (√2) is an irrational number
Hence our assumption that " 4−3√2 = (a/b) is a rational number" is incorrect.
Thus  4−3√2 is an irrational number.
hope it will help u...

Answer 2

Answer:

Let (4-5√2) be a rational number

then,there exist co prime a and b such that

(4-5√2)= a/b

5√2= a-4b/b

√2= a-4b/5b

since a and b are integers so a-4b/5b is rational

thus √2 is also rational but this contradict the fact that √2 is irrational. so our assumption is incorrect

Hence ,(4-5√2) is irrational

if there is any mistake in my solution kindly mention in comments and I hope it will help you