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,

Both numbers are positive, so the equality is true if, and only if its squares are equal. By squaring both sides we get
$$14+4\sqrt{10}-2\sqrt{(14+4\sqrt10)(14-4\sqrt{10})}+14-4\sqrt{10} = 4^2$$
$$14-2\sqrt{(14^2-16 \cdot 10)}+14 = 16$$
$$28-12 = 16$$
$$16 = 16$$
