Question

Show that 4√11 is an irrational number.​

Answer 1

Answer:

A similar approach

Step-by-step explanation:

By the method of contradiction..

Let √11 be rational , then there should exist √11=p/q ,where p & q are coprime and q≠0(by the definition of rational number). So,

√11= p/q

On squaring both side, we get,

11= p²/q² or,

11q² = p². …………….eqñ (i)

Since , 11q² = p² so ,11 divides p² & 11 divides p

Let 11 divides p for some integer c ,

so ,

p= 11c

On putting this value in eqñ(i) we get,

11q²= 121p²

or, q²= 11p²

So, 11 divides q² for p²

Therefore 11 divides q.

So we get 11 as a common factor of p & q but we assumpt that p & q are coprime so it contradicts our statement. Our supposition is wrong and √11 is irrational.

Answer 2

$\huge\star{\red{\underline{\mathfrak{Answer}}}}$

Let us assume that $4 \sqrt{11}$ is a rational number.

So, $4 \sqrt{11} = \frac{p}{q}$, where p and q are co primes.

$\implies{ \sqrt{11} = \frac{p}{4q}}$

The RHS is rational whereas the LHS is irrational.

Therefore, our assumption is contradicted.

So,

$4 \sqrt{11}$

is irrational.

Hence Proved.

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NOTE :- In this type question the number given should be assumed as a rational number.

Then we should prove that LHS is not equal to RHS.

Eventually, the prove gets completed.

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