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During some calculations, I came across a fraction of this kind :

$$\frac{\sqrt{2}+2}{\sqrt{2}+1}$$

Which should be equal to $\sqrt{2}$.

I am surprised to see that Sage can't simplify this fraction with simplify_full :

( (sqrt(2)+2)/(sqrt(2)+1) ).simplify_full()

returns the same. Just to be sure:

bool( (sqrt(2)+2)/(sqrt(2)+1) == sqrt(2) )

returns true

Am I missing a simplification option ? How can I get Sage to simplify this fraction ?

During some calculations, I came across a fraction of this kind :

$$\frac{\sqrt{2}+2}{\sqrt{2}+1}$$

Which should be equal to $\sqrt{2}$.

I am surprised to see that Sage can't simplify this fraction with simplify_full :

( (sqrt(2)+2)/(sqrt(2)+1) ).simplify_full()

returns the same. Just to be sure:

bool( (sqrt(2)+2)/(sqrt(2)+1) == sqrt(2) )

returns true

Am I missing a simplification option ? How can I get Sage to simplify this fraction ?

f = 3*(x^4+4*sqrt(3)*(x^2+6)*sqrt(x^2+3)+24*x^2+72)/(sqrt(3)*(x^5+24*x^3+72*x)+12*(x^3+6*x)*sqrt(x^2+3))

During some calculations, I came across a fraction of this kind :

$$\frac{\sqrt{2}+2}{\sqrt{2}+1}$$

Which should be equal to $\sqrt{2}$.

I am surprised to see that Sage can't simplify this fraction with simplify_full :

( (sqrt(2)+2)/(sqrt(2)+1) ).simplify_full()

returns the same. Just to be sure:

bool( (sqrt(2)+2)/(sqrt(2)+1) == sqrt(2) )

returns true

Am I missing a simplification option ? How can I get Sage to simplify this fraction ?

To clarify, the original expression I encountered was this one :

$$\frac{3(x^4+4\sqrt{3}(x^2+6)\sqrt{x^2+3}+24x^2+72)}{\sqrt{3}(x^5+24x^3+72b)+12(x^3+6b)\sqrt{x^2+3}}$$ $$\frac{3(x^4+4\sqrt{3}(x^2+6)\sqrt{x^2+3}+24x^2+72)}{\sqrt{3}(x^5+24x^3+72x)+12(x^3+6x)\sqrt{x^2+3}}$$

which is equal to $\frac{\sqrt{3}}{x}$. Sage can show the equality, but cannot simplify the expression (but maybe it's normal, this is not as trivial as the first example...). Substituting $x=1$ in this formula give something very similar to the expression above.

It can be obtained with:

f = 3*(x^4+4*sqrt(3)*(x^2+6)*sqrt(x^2+3)+24*x^2+72)/(sqrt(3)*(x^5+24*x^3+72*x)+12*(x^3+6*x)*sqrt(x^2+3))