# Simplifying surds upon surds

1. Jan 23, 2009

### Mentallic

For some surds inside of surds, they can be converted into a more simple form:

$$\sqrt{a+b\sqrt{c}}=e+f\sqrt{g}$$

Such as: $$\sqrt{11-6\sqrt{2}}=3-\sqrt{2}$$

However, there are some that cannot be simplified into this form (as far as I know).

Such as: $$\sqrt{3+\sqrt{7}}$$

I am curious to know if there is fast method in realizing whether these types of equations can be simplified.
My only way of knowing so far is as follows:

To see if $$\sqrt{31+12\sqrt{3}}$$ can be simplified, first I let it be equal to some general simplified form:

$$\sqrt{31+12\sqrt{3}}=a+b\sqrt{3}$$

squaring both sides:

$$a^2+3b^2+2\sqrt{3}ab=31+12\sqrt{3}$$

Therefore, $$a^2+3b^2=31$$ (1) and
$$2\sqrt{3}ab=12\sqrt{3}$$ (2)

Making a or b the subject in (2)
$$b=\frac{6}{a}$$

Substituting into (1)

$$a^2+3(\frac{36}{a^2})=31$$

$$a^4-31a^2+108=0$$

Now we have a quadratic in $$a^2$$. I will now know from the quadratic formula if this expression can be simplified or not by looking at the discriminant. If it is a perfect square, then it can be simplified, else, it cannot be.

2. Jan 26, 2009

3. Feb 2, 2009

### Mentallic

Given $$\sqrt{A \pm B}$$ where $$A,B$$ all reals, $$A+B>0$$
Checking to see if $$\sqrt{A^2-B^2}$$ is a rational number will give me the indication whether to pursue the simplified answer.