Question

Prove that `-7 - 2sqrt3` is an irrational number, given that `sqrt3` is an irrational number.

Answer 1

Let us assume that  `-7 - 2sqrt3` is a rational number.

Then, it will be the form `a/b`, where a, b are coprime integers and b ≠ 0.

Now, `-7 - 2sqrt3 = a/b`

On rearranging, we get

 `-7 -a/b = 2sqrt3`

`-7/2 - a/(2b) = sqrt3`

Since, `7/2 and a/(2b)` are rational.

So, their difference will be rational.

∴ `sqrt3` is rational.

But given that `sqrt3` is an irrational number.

So, this contradicts the fact that `sqrt3` is rational.

Therefore, our assumption is wrong.

Hence, `-7 - 2sqrt3` is irrational.