Question

Prove that √3 + √5 is an irrational number.​

Answer 1

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Prove that √3 + √5 is an irrational number.

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✒ Let √3+√5 be a rational number.

A rational number can be written in the form of p/q where p,q are integers.

√3+√5 = p/q

√3 = p/q-√5

Squaring on both sides,

(√3)² = (p/q-√5)²

3 = p²/q²+√5²-2(p/q)(√5)

√5×2p/q = p²/q²+5-3

√5 = (p²+2q²)/q² × q/2p

√5 = (p²+2q²)/2pq

p,q are integers then (p²+2q²)/2pq is a rational number.

Then √5 is also a rational number.

But this contradicts the fact that √5 is an irrational number.

So,our supposition is false.

Therefore, √3+√5 is an irrational number.

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Answer 2

$\huge\underline \bold\pink {Answer}$

Let √3+√5 be a rational number.

A rational number can be written in the form of p/q where p,q are integers.

√3+√5 = p/q

√3 = p/q-√5

Squaring on both sides,

(√3)² = (p/q-√5)²

3 = p²/q²+√5²-2(p/q)(√5)

√5×2p/q = p²/q²+5-3

√5 = (p²+2q²)/q² × q/2p

√5 = (p²+2q²)/2pq

p,q are integers then (p²+2q²)/2pq is a rational number.

Then √5 is also a rational number.

But this contradicts the fact that √5 is an irrational number.

So,our supposition is false.

Therefore, √3+√5 is an irrational number.