Question

if root11 is irrational, prove that 2+3 root11/3 is irrational

Answer 1

Given:

• √11 is a Irrational number.

To Prove:

• 2+3√11/3 is a Irrational number.

Proof:

Given that √11 is a Irrational number and we are asked to Prove that 2+3√11/3 is also a Irrational number.

Now , we know that :-

• Sum of rational number and a Irrational number is a Irrational number.
• Division of rational number and a Irrational number is a Irrational number.
• Product of rational and a Irrational number is Irrational number

Let's Prove it now ,

Firstly it is given in the numerator 2+3√11 .Here look at 3√11 , so it is a product of Irrational and a rational number so it must be a Irrational number.

Again it is 2+3√11 , now we know that the sum of z rational and a Irrational Number is also a Irrational Number .Hence the numerator is Irrational .

In denominator it is given 3 , which is a rational number and we already proved that the numerator is Irrational .And we know division of a rational and a Irrational Number is also a Irrational Number .

Hence the whole Number is Irrational .

Hence Proved :)

Answer 2

• Firstly we assume that √11 is a rational number.
• A rational number can be written in the form of p/q, where q ≠ 0 and p , q are positive number.
• √11 = p/q ….( Where p & q are co prime number )
• Square both sides, we get
• 11 = p²/q²
• 11 q² = p² ….(a)
• p² is divisible by 11. So, p will also divisible by 11
• Let p = 11 r ( Where r is any positive integer )
• Squaring both sides
• p² = 121r²
• Putting this value in eqn(a)
• 11 q² = 121 r²
• q² = 11 r²
• q² is divisible by 11. So, q will also divisible by 11. Since, p and q both are divisible by same number 11. So, they are not co-prime .
• Hence Our assumption is Wrong. Therefore, √11 is an irrational number .