Question

Prove that √5, √6 and √7 are irrational numbers​

Answer 1

Answer:

Assume that √5 is rational

√5= p /q where p and q are co-primes

P = √5q

Squaring both sides

P²  =  (√5q)²

P² = 5q² -----------------(1)

Now p² is divisible by 5 so that by Theorem 1.2 p is also divisible by 5

So,     p = 5r        (where r is any positive integer) --------------(2)

Putting value of eq. (2) in (1) we get,

25r² = 5q²

Now on dividing from 5 on both sides we get,

5r² = q²

So we can conclude that p and q both have common factor 5 so they are not co-prime.

This problem erosion due to wrong assumption that √5 is rational. So, √5 is irrational.

Similarly , we can prove that √6 and √7 as irrational numbers.

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