Question

Prove that 5+2 root 3 is irtational

Answer 1

Hey friend,

Here is your answer:

To prove:

5 +2√3 is irrational.

Assumption:

Let us assume 5 + 2√3 to be " a " and it is rational.

Proof,

As, 5 + 2√3 Is rational it can be written in the form of p/q where q ≠ 0.

(p , q are coprime)

Then,

$5+2 \sqrt{3} = \frac{p}{q}$

→$2 \sqrt{3} = \frac{p}{q} - 5$

→$2 \sqrt{3} = \frac{p-5q}{2}$

→ $\sqrt{3} = \frac{p-5q}{2q}$

We know that ,

√3 is irrational .

And, $( \frac{p-5q}{2q})$ is rational.

We know that ,

Irrational ≠ rational..

So, we contradict the statement that 5 + 2 √ 3 is rational.

Therefore 5 + 2√3 is an irrational number

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Hope my answer is helpful to you.

Answer 2

Answer:

∛5 + 2 = ∛7 = irrational