Question

Prove that 2+√5/3 is irrational...............Please please it is very urgent and please answer correct please!!!!​

Answer 1

equate 2+√5/3 to a/b now transpose the numericals

you get

√5 =(a-2b/b)3

√5=3a-6b/b

now as we know root 5 is an irrational no. so 2+√5/3 is also an irrational no.

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Answer 2

Answer:

first prove that √5 is irrational

let us assume√5 as rational

=> √5= p/q (where p and q are co-primes)

(squaring on both sides)

(√5)^2 = (p/q)^2

${p}^{2} \: \: \: \: \: \\ - \: = 5 \\ {q}^{2} \: \: \: \: \: \: \: \:$

=> p^2 = 5q^2

therefore 5 is a factor of p^2

=> 5 is a factor of p

so p^2 can be written as (5c)^2

(5c)^2 = 5q^2

25c^2 = 5q^2

( divided both sides with 5)

after dividing

5c^2 = q^2

therefore 5 is a factor of q^2

=> 5 is a factor of q

but it is not possible as p and q are co-primes

therefore, our assumption is wrong

=> √5 is irrational

as √5 is irrational 2+√5/3 is also irrational

as any irrational number added or divided to a rational number gives irrational