Question

prove that √5 is irrational and hence show that 7+2√5 is also an irrational number using contradiction method​

Answer 1

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prove that √5 is irrational ??

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Given: √5

We need to prove that √5 is irrational

Proof:

Let us assume that √5 is a rational number.

So it 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...

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show that 7+2√5 is also an irrational number ???

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7 + 2√5 is rational number because 2√5 is irrational number

2√5 is irrational number because √5 is irrational

√5 - 2.23606797749 so √5 is irrational so therefore 7 + 2√5 is irrational number ...

Answer 2

$\:$

yaa she is right