Question

i) Prove that root 7 +root 5 is an irrational number​

Answer 1

Answer:

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Answer 2

Given : √7 + √5

We need to prove√7 + √5 is an irrational number.

Proof :

Let us assume that √7 + √5 is a rational number.

A rational number can be written in the form of p/q where p & q are integers and q≠0

√7 + √5 = p/q

On squaring both sides we get,

(√7 + √5)² = (p/q)²

√7² + √5² + 2(√5)(√7) = p²/q²

7 + 5 + 2√35 = p²/q²

12 + 2√35 = p²/q²

2√35 = p²/q² – 12

√35 = (p²-12q²)/2q²

• p & q are integers then (p²-12q²)/2q² is a rational number.

• Then √35 is also a rational number.

• But this contradicts the fact that √35 is an irrational number.

• Our assumption is incorrect

√7 + √5 is an irrational number.

Hence proved.