Question

Prove that ${\sqrt{5}}+ {\sqrt{3}}$ is irrational

Answer 1

SOLUTION :  

Let us assume , to the contrary ,that √5 + √ 3 is rational. Then,it will be of the form a/b where a, b are co primes integers and b ≠0.

√5 + √ 3 = a/b

√5 = a/b - √3

On squaring both sides ,

(√5 )² = (a/b - √3)²

5 = (a/b)² + √3² - 2× a/b × √3

[(a-b)² = a² + b² - 2ab]

5 = (a/b)² + 3 - (2× a × √3)/b

5 - 3 = (a/b)² - (2× a × √3)/b

2 = (a/b)²  - (2× a × √3)/b

(2× a × √3)/b = (a/b)² - 2

(2× a × √3)/b = a²/b² - 2

(2a √3)/b =( a² - 2b²)/b²

√3 = (a² - 2b²)/b² × (b/2a)

√3 = (a² - 2b²) /2ab

since, a & b is an integer so,(a² - 2b²) /2ab   is a rational number.  

∴ √3 is rational  

But this contradicts the fact that √3 is an irrational number .

Hence,√5 + √ 3 is an irrational .

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Answer 2

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