I am having trouble with questions like these:
$$\sqrt{x-y\sqrt z}$$
Or as an example:
$$\sqrt{17-12\sqrt2}$$
I cannot see how to go about this. I have tried squaring the expression, which ends up giving $\sqrt{289}-\sqrt{288}$, but I do not know where to go from here.
Can anybody help?
You need to find how $\sqrt{x-y\sqrt{z}}$ could be a perfect square.
We assume
$$x-y\sqrt z=(a-b\sqrt z)^2=a^2+b^2z-2ab\sqrt z$$ where $a,b$ are rational.
Then we need to solve
$$\begin{cases}a^2+b^2z=x,\\2ab=y\end{cases}.$$
Multiplying by $a^2$,
$$a^4-a^2x+a^2b^2z=a^4-a^2x+\frac{y^2}4z=0.$$
This equation has a rational solution if the discriminant
$$\Delta=x^2-y^2z$$
is a perfect square, and one of the roots in $a^2$ is a perfect square
$$\frac{x\pm\sqrt\Delta}2.$$
With the given example,
$$\Delta=1,\\\frac{17\pm1}2=8,9.$$