- #1
- 3,802
- 95
For some surds inside of surds, they can be converted into a more simple form:
[tex]\sqrt{a+b\sqrt{c}}=e+f\sqrt{g}[/tex]
Such as: [tex]\sqrt{11-6\sqrt{2}}=3-\sqrt{2}[/tex]
However, there are some that cannot be simplified into this form (as far as I know).
Such as: [tex]\sqrt{3+\sqrt{7}}[/tex]
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 [tex]\sqrt{31+12\sqrt{3}}[/tex] can be simplified, first I let it be equal to some general simplified form:
[tex]\sqrt{31+12\sqrt{3}}=a+b\sqrt{3}[/tex]
squaring both sides:
[tex]a^2+3b^2+2\sqrt{3}ab=31+12\sqrt{3}[/tex]
Therefore, [tex]a^2+3b^2=31[/tex] (1) and
[tex]2\sqrt{3}ab=12\sqrt{3}[/tex] (2)
Making a or b the subject in (2)
[tex]b=\frac{6}{a}[/tex]
Substituting into (1)
[tex]a^2+3(\frac{36}{a^2})=31[/tex]
[tex]a^4-31a^2+108=0[/tex]
Now we have a quadratic in [tex]a^2[/tex]. 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.
[tex]\sqrt{a+b\sqrt{c}}=e+f\sqrt{g}[/tex]
Such as: [tex]\sqrt{11-6\sqrt{2}}=3-\sqrt{2}[/tex]
However, there are some that cannot be simplified into this form (as far as I know).
Such as: [tex]\sqrt{3+\sqrt{7}}[/tex]
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 [tex]\sqrt{31+12\sqrt{3}}[/tex] can be simplified, first I let it be equal to some general simplified form:
[tex]\sqrt{31+12\sqrt{3}}=a+b\sqrt{3}[/tex]
squaring both sides:
[tex]a^2+3b^2+2\sqrt{3}ab=31+12\sqrt{3}[/tex]
Therefore, [tex]a^2+3b^2=31[/tex] (1) and
[tex]2\sqrt{3}ab=12\sqrt{3}[/tex] (2)
Making a or b the subject in (2)
[tex]b=\frac{6}{a}[/tex]
Substituting into (1)
[tex]a^2+3(\frac{36}{a^2})=31[/tex]
[tex]a^4-31a^2+108=0[/tex]
Now we have a quadratic in [tex]a^2[/tex]. 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.