Question

prove that 5+2underroot3 is a irrational number.

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Answer 1

Answer:

Hey friend, here.

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}5−2 3 = qp

→ -2 \sqrt{3} = \frac{p}{q} - 5−2 3 = qp −5

→ - 2 \sqrt{3} = \frac{p-5q}{2}−2 3 = 2p−5q

→ \sqrt{3} = - ( \frac{p-5q}{2q}) 3

=−( 2qp−5q )

We know that ,

√3 is irrational .

And, - ( \frac{p-5q}{2q})−(

2qp−5q ) 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:

Let, assume that , 5 + 2√3 is an rational number

5 + 2√3 = a/b

2√3 = a/b - 5

√3 = (a - 5b)/2b

Here , √3 is an irrational number but (a - 5b)/2b is rational number

Since , irrational ≠ rational

This is a contradiction , thus our assumptions is wrong.

Therefore , 5+2√3 is irrational number.

hope this will help you.