Question

Prove that 15 - 7√2 is irrational , given that √2 is irrational.​

Answer 1

Answer:

Let us assume the contrary.

i.e; 5 + 3√2 is rational

∴ 5 + 3√2 = ab, where ‘a’ and ‘b’ are coprime integers and b ≠ 0

3√2 = ab – 5

3√2 = a−5bb

Or √2 = a−5b3b

Because ‘a’ and ‘b’ are integers a−5b3b is rational

That contradicts the fact that √2 is irrational.

The contradiction is because of the incorrect assumption that (5 + 3√2) is rational.

So, 5 + 3√2 is irrational.

Answer 2

Let 15+17√2 is a rational number

we know that any rational no. it is in the form of p/q ,where pand q are co-prime no.

15+17√2=p/q

17√2=p/q -15

√2=p-15q/17q ...........(I)

here, p and q are some integers

therefore, p-15q/17q is a rational no.

but, √2 is irrational no.

it contradicts our supposition

=> 6+√2 is irrational .............(hence proved)