Question

5+√3is irrational number

Answer 1

Explanation:

To prove : 5+√3 is irrational

Proof:

Let us assume that 5 +√3 is rational.

Let ,

5 + √3 = r , where "r" is rational

5 + r = √3

Here,

LHS is purely rational.But,on the other hand ,RHS is irrational.

This leads to a contradiction.

Hence,5+√3 is irrational

Answer 2

Answer:

Let us assume 5+ √3 is rational

So, 5+ √3 = p/q, where p & q are co primes

Then, 5 = (p/q)-√3

Squaring both sides,

⇒25=( p²/q²)+3 - 2√3p/q

⇒2√3p/q=( p²/q²) - 22

⇒√3=(p²-22q²)q/2pq²

⇒ √3 = (p²-22q²)/2pq

Since, p and q are integers, (p²-22q²)/2pq is rational

∴ √3 is rational

But this contradicts the fact √3 is irrational. The contradiction arises when we assume that(5+ √3) is rational.

Hence, (5+ √3) is irrational. [Proved]

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