Question

prove that (root3-root2) is irrational​

Answer 1

Step-by-step explanation:

If possible let root 3 is a rational number.

Therefore, root 3 = a/b , where a and b are co- prime integers .

that is, root 3 =a/b

Now,

squaring both side , we have ,

( root 3 )^2 = (a/b)^2

or, 3= a^2/ b^2.

or, a^2 = 3b^2

or, we can say.,

a^2 is divisible by 3

a is divisible by 3.

Again,

let, a=3c , where c is an integer .

a= 3c

squaring both sides

(a)^2= (3c)^2

or, a^2 = 9c ^2

or, 3b^2 = 9c^2 [ a^2 = 3b^2].

or, b^2= 3c^2

or, we can say,

b ^2 is divisible by 3

b is divisible by 3

Thus, a and b has no common factor other than 3.

also, a and b are divisible by 3.

There fore, our assumption is wrong.

Hence, root 3 is an irrational number.

In the same way find root 2 .