How to simplify surd expressions such as this?

Question

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?

$\sqrt{17-12\sqrt{2}}=\sqrt{17-2\sqrt{9}\sqrt{8}}=\sqrt{(3-2\sqrt{2})^2}=3-2\sqrt{2}$ — , Sep 16 '20 at 17:34
Does this answer your question? Simplifying $\sqrt[4]{161-72 \sqrt{5}}$ This includes two general approaches for such problems. — kingW3, Sep 16 '20 at 17:36
The problem you have suggested I look at gives me a good start - however I come upon a roadblock, because I get the expressions a^2+b^2=17, and 2ab = 12. a^2 + 2b^2 can only be factorised into (a+b(√2))*(a-b(√2)), which doesn't help me when 2ab = 12. — oscar6721, Sep 16 '20 at 17:51

score 1 · Answer 1 · answered Sep 16 '20 at 17:51

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,

$$\Delta=1,\\\frac{17\pm1}2=8,9.$$

How to simplify surd expressions such as this?

1 Answers1