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