<s>[INST] How do I prove that $\sqrt{20+\sqrt{20+\sqrt{20}}}-\sqrt{20-\sqrt{20-\sqrt{20}}} \approx 1$ How do I prove that
$$\sqrt{20+\sqrt{20+\sqrt{20}}}-\sqrt{20-\sqrt{20-\sqrt{20}}} \approx 1$$
without using the calculator?
 [/INST] By binomial formulas and cancellation, you get that
\begin{align}
...&=\frac{\sqrt{20+\sqrt{20}}+\sqrt{20-\sqrt{20}}}{\sqrt{20+\sqrt{20+\sqrt{20}}}+\sqrt{20-\sqrt{20-\sqrt{20}}}}
\\&=\frac{
  \sqrt{5+\sqrt{\frac54}}+\sqrt{5-\sqrt{\frac54}}
  }{
  \sqrt{5+\sqrt{\frac54+\sqrt{\frac5{64}}}}+\sqrt{5-\sqrt{\frac54-\sqrt{\frac5{64}}}}
  }
\end{align}
As one can see, the denominator is a small perturbation of the numerator, so that the quotient will be close to $1$.
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