Question

If is √ab an irrational number prove that (√a+√b) is an irrational number.​

Answer 1

Let us suppose that √a + √b is a rational number.

So, we can write √a + √b as

√a + √b = p/q .........1 where q ≠ 0

Now, squaring on both side, we get

      (√a + √b)2 = (p/q)2

=> a + b + 2*√a * √b = p2 /q2

=> a + b + 2*√(ab) = p2 /q2

=> 2*√(ab) = p2 /q2 - (a + b)  ............2

Since, √a + √b are rational, So a and b are also rational.

So, RHS of equation 2 is a rational number.

But it is given that √(ab) is an irrational number.

So, LHS of equation 2 is an irrational number.

which contradict our assumption.

Answer 2

 

Step-by-step explanation: As we know that any number in the root is the half of the number which is in the root .

So, the mostly number in the root are does not be written p/q form which is the actual form of rational number.

Therefore , root ab and root a+root b both are the irrational numbers.