Question

prove that root 3 -root 2 is an irrational number​

Answer 1

Question

prove that √3-√2 is an irrational number

$\sf\huge\bold{\underline{\underline{{Solution}}}}$

First we have to assume that √3-√2 is a rational number.

Let us assume that √3-√2 is a rational number say p

Our equation : √3-√2=p

Squaring both sides

we obtained : (√3-√2)²=p²

Identity used : (a-b)²=a²+b²-2ab

=>(√3)²+(√2)²-2(√3)(√2)=p²

=>3+2-2√6=p²

=>5-2√6=p²

=>-2√6=p²-5

=>√6= -( p²-5) / 2

Here,we see that √6 is an irrational number and -(p²-5)/2 is in the form of a p/q which is a rational number.

Since,rational number can't be equal to an irrational number.

Hence,our assumption of √3-√2 is a rational number is wrong.

Therefore,√3-√2 is an irrational number