Question

prove that root 5 is an irrational number
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Answer 1

Rational number:a number which can be written in the form of p/q where p and q are integers

Irrational number:a number which cannot be written in the form of p/q where p and q are integers

root 5 cannot be written in the form of p/q

therefore root 5 is an irrational number

Answer 2

$\huge \underline \mathbb {SOLUTION:-}$

Given: √5

We need to prove that√5 is irrational

Proof:

Let us assume that √5 is a rational number.

Sp it t can be expressed in the form p/q where p,q are co-prime integers and q≠0

⇒√5=p/q

On squaring both the sides we get,

⇒5=p²/q²

⇒5q²=p² —————–(i)

p²/5=q²

So 5 divides p

p is a multiple of 5

⇒p=5m

⇒p²=25m² ————-(ii)

From equations (i) and (ii), we get,

5q²=25m²

⇒q²=5m²

⇒q² is a multiple of 5

⇒q is a multiple of 5

• Hence, p,q have a common factor 5. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number √5 is an irrational number

Hence proved ❤️❤️

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