Question

prove that 3√ 2 is an irrational number

Answer 1

To prove that 3+root2 is an irrational number

lets take the opposite i.e 3+root2 is a rational number

hence 3+root2 can be written in the form a/b

hence 3+root2 = a/b

          root2 = 1/3 x a/b

          root2 = a/3b

here a/3b is rational and root2 is irrational

as irrational cannot be equal to rational 

3+root2 is irrational

Answer 2

HEY MATE!

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HERE IS YOUR ANSWER:

To prove: 3 root 2 is irrational.

For proving it let us first prove that root 2 is irrational.

Let us assume to that root2 is rational, and root2=a/b where a and b are two co-primes that have no common factor other than 1.

now, root2=a/b

squaring both sides

we get, 2=a^2/b^2

2b^2=a^2

This shows us that a^2is divisible by 2

and hence a is divisible by 2.

now, let us take a=2c for any positive integer c.

putting the value of a=2c

we get, 2b^2=(2c)^2

i.e 2b^2=4c^2

now, b^2=2c^2

This shows us that b^2 is divisible by 2 and hence b is divisible by 2.

Now, we get that a and b both have a common factor 2.

Hence, it contradicts our assumption that a and b are co-primes .

This shows that our assumption that root2 is rational is wrong.

Hence root2 is irrational.

Now, we need to prove that 3root2 is irrational.

let us assume that 3 root2 is rational where 3 root2=a/b where a and b are two co-primes.

So, 3 root2=a/b

root2=a/b-3

root2=a-3b/b

now, a-3b/b is irrational as we proved that root2 is irrational.

But this contradicts our assumption that 3 root2 is rational.

Thus, our assumption was wrong and 3 root2 is an irrational number.

Hence proved.

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HOPE IT HELPED^_^