Question

Prove that underroot 2+3/underroot 2 is an irrational number

Answer 1

Let us 2+3/√2 to be a rational.

=>2+3/√2=p/q ,where p and q are natural numbers and q≠0

=>3/√2=p/q-2

=>3/√2=(p-2q)/q

=>1/√2=(p-2q)/3q

=>√2=3q/p-2q

It is known that √2 is an irrational

=>LHS≠RHS

Thus,2+3/√2 is not a rational and is an irrational

Hence,proved

Other method:

If a rational and an irrational are divided,the solution is an irrational

So,3/√2 is an irrational

If a rational and an irrational are added,the sum is an irrational.

So,2+3/√2 is an irrational

Hence,2+3/√2 is an irrational

Answer 2

Answer:

Let us 2+3/√2 to be a rational.

=>2+3/√2=p/q ,where p and q are natural numbers and q≠0

=>3/√2=p/q-2

=>3/√2=(p-2q)/q

=>1/√2=(p-2q)/3q

=>√2=3q/p-2q

It is known that √2 is an irrational

=>LHS≠RHS

Thus,2+3/√2 is not a rational and is an irrational

Hence,proved

Other method:

If a rational and an irrational are divided,the solution is an irrational

So,3/√2 is an irrational

If a rational and an irrational are added,the sum is an irrational.

So,2+3/√2 is an irrational

Hence,2+3/√2 is an irrational

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Step-by-step explanation: