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Question Number 192387 by cortano12 last updated on 16/May/23

$$\mathrm{Simplify}$$$$\sqrt{2(1+\sqrt{4+{\left(\frac{{2017}^4-1}{{2017}^2}\right)}^2})}$$$$\mathrm{is}....$$

Answered by a.lgnaoui last updated on 16/May/23

$$\sqrt{4+{\left(\frac{{2017}^4-1}{{2017}^2}\right)}^2}=\sqrt{4+{({2017}^2-\frac{1}{{2017}^2})}^2}$$$$=\sqrt{4+{(2017-\frac{1}{2017})}^2{(2017+\frac{1}{2017})}^2}$$$$=\sqrt{4+\left[({2017}^2+\frac{1}{{2017}^2}-2)({2017}^2+\frac{1}{{217}^2}+2)\right]}=$$$$=\sqrt{{({2017}^2+\frac{1}{{2017}^2})}^2}={2017}^2+\frac{1}{{2017}^2}$$$$\Rightarrow \sqrt{2(1+{(4+\sqrt{\frac{{2017}^4-1}{{2017}^2}})}^2}=$$$$\sqrt{2(1+{2017}^2+\frac{1}{{2017}^2})}=\sqrt{2+2[{(2017+\frac{1}{2017})}^2-2]}$$$$=\left[\sqrt{1+{(2017+\frac{1}{2017})}^2-1}\right]\sqrt{2}$$$$=(2017+\frac{1}{2017})\sqrt{2}$$

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