Question

5-2root 3 is irrational number

Answer 1

Yeah, 5 - 2 √3 is an irrational number.

Proof: Let 5 - 2 √3 is a rational number equal to a .

( 5 - 2 √3 ) = a

Squaring both sides,

( 5 - 2√3 ) ^ 2 = a ^ 2

5 ^ 2 + ( 2 √3 ) ^2 - 2 * 5 * ( 2√3) = a ^ 2

25 + 12 - 20 √3 = a ^ 2

37 - 20√3 = a ^ 2

20√3 = 37 - a ^ 2

√3 = ( 37 - a ^ 2 ) / 20.

Since, a is a rational number so ( 37 - a ^ 2 ) / 20 is also a rational number and it is equal to √3,but a rational number can't be equal to an irrational number.So it contradicts our assumption that 5 - 2 √3 is a rational number.Hence, it is an irrational number.