1. Prove that √3 is irrational and hence show that 2+√3/5is irrational.

Answers 1

Step-by-step explanation:

Let us consider 5 be a rational number, then

5=p/q, where ‘p’ and ‘q’ are integers, q=0 and p, q have no common factors (except 1).

So,

5=p2/q2

p2=5q2 …. (1)

As we know, ‘5’ divides 5q2, so ‘5’ divides p2 as well. Hence, ‘5’ is prime.

So 5 divides p

Now, let p=5k, where ‘k’ is an integer

Square on both sides, we get

p2=25k2

5q2=25k2 [Since, p2=5q2, from equation (1)]

q2=5k2

As we know, ‘5’ divides 5k2, so ‘5’ divides q2 as well. But ‘5’ is prime.

So 5 divides q

Thus, p and q have a common factor of 5. This statement contradicts that ‘p’ and ‘q’ has no common factors (except 1).

We can say that 5 is not a rational number.

Hope it helps

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