Question

Prove that root 2 minus root 3 is irrational no. ?

Answer 1

let that the given number as irrational √2-3
so..use the following steps to solve this problem
step1-let √2-3 =a/b
          a,b∈z
          b≠0
step2-squaring on both sides
        ( √2-3)²=(a/b)²
         4-9      =a²/b²
          -5        =a²/b²...................-5b²=a²........eq-1
        case2:
            c=a/b
             s.o.b.s
             c²=a²/b²
             c²b²=a²...............eq.2
            from eq1,2
            c²b²=-5b²
             it is clear that b is a integer that is rational
              soo....hence proved that given statement is irrational in number
             it contradicts the fact that is irrational..

          
        

Answer 2

Hey!

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Let √2 - √3 be rational,

√2 - √3 = a/b

√2 = a/b + √3

Squaring both sides-

( √2 ) ^2 = ( a/b + √3) ^2

[Using identity, (a+b)^2 = a^2 + 2ab + b^2]

2 = (a/b)^2 + (√3)^2 + 2 × a/b × √3

2 = a^2 / b^2 + 3 + 2 (a/b) (√3)

2 - 3 = a^2 / b^2 + 2 (a/b) (√3)

-1 = a^2/b^2 + 2(a/b) (√3)

-2 (a/b) (√3) = a^2/b^2 + 1

√3 × -2a/b = a^2/b^2 + 1

√3 = (a^2/b^2 + 1) b/-2a

√3 = ( a^2 + b^2 / -2ab)

Now,

a^2 + b^2/ -2ab is rational,

√3 = a^2 + b^2/ -2ab is also rational then, but this contradicts the fact that √3 is irrational.

Thus, as √3 is irrational, so, √3 = a^2 + b^2/ -2ab is irrational too.

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Hope it helps...!!!