Question

Root 20 is irrational Prove it ​

Answer 1

Answer: 20 is not a perfect square, so its square root is irrational.

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Answer 2

Answer:

Root 20 = $2\sqrt{5}$

Let us consider root 5 as rational such that it can be expressed as p/q where p and q have no common factor

p/q = root 5

q root 5 = p

5$q^{2}$ = $p^{2}$                    (1)

Thus, $p^{2}$ is divisible by 5 and so we understand that even p is divisible by 5

so we can write p as 5x for some integer x

substituting for p in (1), we get

$25x^2$ = 5$q^{2}$

so $q^{2} = 5x^{2}$

means that q2 is divisible by 5 so even q is divisible by 5

So we find that both p and q are divisible by 5 but earlier we had considered p and q have no common factor. Thus, we find root 5 cannot be represented as p/q where p and q have no common factor so root 5 is irrational and thus, we can say $2\sqrt{5}$ is also irrational

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