Question

$prove \: \sqrt{5} - \sqrt{3} \: \: is \: irrational$
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Answer 1

let us assume that ✓3 is a rational number such that ✓3= p/q where p and q are positive integers and hcf(p,q)=1

=> ✓3 = p/q

=> (✓3)^2 = p^2/q^2 ( squaring both

sides)

=> 3 = p^2/q^2

=> q^2 = p^2/3

since, p^2 is divisible by 3

therefore, p is also divisible 3

therefore, p = 3r

=> q^2 = (3r)^2/3

=> q^2 = 9r^2/3

=> q^2 = 3r^2

=> 3r^2 = q^2

=> r^2 = q^2/3

since, q^2 is divisible by 3

therefore, q is also divisible 3

therefore, p and q both have 3 as their common factor.

but contradicts the fact that p and q have no common factor other than 1 ..

so, out assumption is wrong

therefore, ✓3 is an irrational number..

hope you find answer

Answer 2

Answer:

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