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Prove-that-numbers-3-sqrt-5-irrational - Mathematics

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Prove that of the numbers `3/sqrt(5)` is irrational:

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Solution

Let `3/sqrt(5)` be rational.

∴ +`1/3 ×  3/sqrt(5) = 1/sqrt(5)` = rational    [∵Product of two rational is rational]
This contradicts the fact that `1/sqrt(5)` is irrational.
∴` (1 × sqrt(5))/(sqrt(5) × sqrt(5)` = `1/5 sqrt(5)`

So, if  `1/sqrt(5)`  is irrational, then  `1/5 sqrt(5)`  is rational

`∴ 5 xx 1/5 sqrt(5) = sqrt(5)=` ration   [ ∴ Product of two rationalis rational ]

Hence `1/sqrt(5)` is irratonal
The contradiction arises by assuming  `3/sqrt(5)`  is rational.

Hence, `3/sqrt(5)`  is irrational.

Concept: Concept of Irrational Numbers
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