Question

prove that 2+3√3 is an irrational number when it is given that √3 is an irrational number

Answer 1

let 2 + 3√3 is an rational number.

2 + 3√3 = p/q

3√3 = p/q - 2

3√3 = p - 2q/q

√3 = p - 2q/3q

Here we see that,

p - 2q/3q is rational.

And we divide when rational number than the number is rational but here it is irrational.

Given √3 is an irrational.

So, our assumption is wrong

2 + 3√3 is an irrational number.

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Answer 2

Let 2+3√3 is rational number

So

2+3√3=p/q

Where p&q are co-prime number

3√3=p/q-2

√3=p/3q-2.

p/3q-2. is rational number because p&q are co-prime number.

But √3 is irrational numbers.

So our assumption is wrong

Then:-

2+3√3 is an irrational number...

Hence proved

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