<s>[INST] How would you show that $\sqrt{14+4\sqrt{10}} - \sqrt{14-4\sqrt{10}} = 4$? I've recently seen a Highschool problem and I was wondering, how would you show that
$$\sqrt{14+4\sqrt{10}} - \sqrt{14-4\sqrt{10}} = 4$$
Thank you for your time,
 [/INST] As an alternative way note that $$\eqalign{\sqrt{14+4\sqrt{10}} - \sqrt{14-4\sqrt{10}} &= \sqrt{\sqrt{10}^2+2\cdot2\sqrt{10}+2^2} - \sqrt{\sqrt{10}^2-2\cdot2\sqrt{10}+2^2}},$$ then use the identity $a^2\pm2ab+b^2=(a\pm b)^2$, and simplify.
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