Question

Root 3 minus root 5 is irrational number

Answer 1

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Answer:

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Step-by-step explanation:

$3 - \sqrt{5 } \: is \: an \: irrational \: no.$

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Answer 2

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Prove that √3 - √5 is an irrational number.

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Let, us prove it by the method of contradiction.

Let us assume that

√3 + √5 is a rational number then, it could be represented in the form p/q where q is not equal to zero, and p and q are co-prime integers.

so,

$\sqrt{3} + \sqrt{5} = \frac{p}{q} \\ \\ \sqrt{3} = \frac{p}{q} - \sqrt{5} \\ \\ squaring \: both \: sides \: we \: will \: get \\ \\ 3 = \frac{ {p}^{2} }{ {q}^{2} } + 5 - \frac{2 \sqrt{5}p }{q} \\ \\ \frac{2 \sqrt{5} p}{q} = \frac{ {p}^{2} }{ {q}^{2} } + 5 - 3 \\ \\ \frac{2 \sqrt{5}p }{q} = \frac{ {p}^{2} + 2 {q}^{2} }{ {q}^{2} } \\ \\ 2 \sqrt{5} = \frac{ {p}^{2} + 2 {q}^{2} }{pq} \\ \\ \sqrt{5} = \frac{ {p}^{2} + 2 {q}^{2} }{2pq}$

Here,

LHS is irrational because √5 is an irrational number.

but

RHS is irrational having integers

it means

LHS is not equal to RHS

this, result is due to our wrong assumption that

√3 + √5 is rational

hence, it causes contradiction

therefore,

√3 + √5 is irrational.

PROVED.