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Question Number 70021 by naka3546 last updated on 30/Sep/19

$$\sqrt[3]{\sqrt{\sqrt{12345689654321233+5333334096\sqrt{12345679}}}-\sqrt{\sqrt{12345689654321233-5333334096\sqrt{12345679}}}}=...$$

Commented by MJS last updated on 30/Sep/19

$$\mathrm{in}\mathrm{this}\mathrm{case}\mathrm{we}\mathrm{can}\mathrm{solve}$$$$\sqrt{a+b\sqrt{c}}=p+q\sqrt{c}\iff a=p^2+{cq}^2\wedge b=2pq$$$$\mathrm{two}\mathrm{times}\mathrm{and}\mathrm{get}$$$$\sqrt{\sqrt{a+b\sqrt{c}}}-\sqrt{\sqrt{a-b\sqrt{c}}}=(4+3\sqrt{c})-(-4+3\sqrt{c})=8$$$$\sqrt[3]{8}=2$$

Commented by ajfour last updated on 30/Sep/19

$$MjSsir,pleasehelpwithQ.69681$$

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