<s>[INST] How to get the value of the root? I have this statement:

If $\frac{\sqrt{7}}{\sqrt{5}+\sqrt{3}} \approx \frac{2}{3}$, Which of
  the following values are the closest to $\sqrt{21}$ ?
A) 68/15 B) 14/3 C) 19/4 D) 55/12 E) 9/2

My development was:
$\frac{\sqrt{7}}{\sqrt{5}+\sqrt{3}}  = \frac{\sqrt{35}-\sqrt{21}}{2} \approx \frac{2}{3}$
My idea was to treat the sign $\approx$ as a sign $=$ and thus eliminate roots and clear $\sqrt21$, with which I have obtained $55/12$ but I do not know if this is correct. 
$\frac{\sqrt{35}-\sqrt{21}}{2} = \frac{2}{3}$
$35 = (\frac{4}{3} +\sqrt{21})^2$
$\sqrt{21} = \frac{110}{9} * \frac{3}{8} = 55/12$
So my doubt is: Can I treat a $\approx$ sign as a $=$ sign to work like an normal equation?
 [/INST] From
$$\left(\frac23\right)^2\approx\left(\frac{\sqrt7}{\sqrt5+\sqrt3}\right)^2$$
we get
$$\frac49\approx\frac{7}{5+3+2\sqrt{15}},$$
hence $\sqrt{15}\approx31/8$.
 Now we have
$$\frac{2\sqrt{3}}3\approx
\frac{\sqrt7\sqrt3}{\sqrt5+\sqrt3}=\frac{\sqrt{21}}{\sqrt5+\sqrt3}.$$
Using the approximate value for $\sqrt{15}$ we arrive in
$$\sqrt{21}\approx\frac{55}{12}.$$
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