Question

State with reason, whether the following is a surd.

`sqrt(sqrt(3) - sqrt(2))`

Answer 1

`sqrt(sqrt(3) - sqrt(2))` is a surd

Reason:

A surd is an irrational number expressed using roots.

Let `x = sqrt(sqrt3 − sqrt2)`. 

If x were rational then x² would be rational. 

But `x^2 = sqrt3 − sqrt2`, so `sqrt3 − sqrt2` would be rational. 

Suppose `sqrt3 − sqrt2` = r (r rational). 

Then `sqrt3 = r + sqrt2` and squaring both sides gives `3 = r^2 + 2 + 2rsqrt2`, so `2rsqrt2 = 1 - r^2` and hence `sqrt2 = (1 - r^2)/(2r)`, which is rational a contradiction (`sqrt(2)` is irrational). 

Therefore, `sqrt3 − sqrt2` is irrational, so x² is irrational, so x is irrational.

Since x is an irrational number given by a root, it is a surd.