Question

given that under root 3 is irrational prove that 5 root 3 minus 2 is an irrational number​

Answer 1

See the attachment hope it helps.

Answer 2

To Prove: (5√3 - 2) is an irrational number .

Given :- √3 is an irrational number .

Solution :-

Let us assume that, (5√3 - 2) is a rational number.

Than, we can write as ,

→ 5√3 - 2 = p/q {where p and q are integers, ‘q’ not = 0.}

→ 5√3 = (p/q) + 2

→ 5√3 = (p + 2q) / q

→ √3 = (p + 2q) / (5q)

Now , as we can see , in RHS , numerator is the sum of 2 integers, which always remains an integer. And denominator is also Multiple of 2 integer, which is not = 0.

Therefore,

→ RHS is a rational number. { As, all the conditions for being a rational number, have been satisfied. }

But,

Given that, LHS is an irrational number.

Hence,

→ LHS ≠ RHS

∴

→ our assumption that (5√3 - 2) is a rational number is wrong.

Hence,

(5√3 - 2) is an irrational number.