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Question Number 97936 by bobhans last updated on 10/Jun/20

$$\mathrm{Find}\mathrm{the}\mathrm{value}\mathrm{of}$$$$\sqrt{45-\sqrt{2000}}+\sqrt{45+\sqrt{2000}}$$

Commented by pranesh last updated on 10/Jun/20

$$findy$$

Answered by john santu last updated on 10/Jun/20

$$\mathrm{let}{\mathrm{z}}_1=\sqrt{45-\sqrt{2000}},{\mathrm{z}}_2=\sqrt{45+\sqrt{2000}}$$$$\iff {\mathrm{z}}_1^2+{\mathrm{z}}_2^2=90\wedge {\mathrm{z}}_1{\mathrm{z}}_2\sqrt{(45-\sqrt{2000})(45+\sqrt{2000})}=5$$$$\mathrm{then}{({\mathrm{z}}_1+{\mathrm{z}}_2)}^2={\mathrm{z}}_1^2+{\mathrm{z}}_2^2+{2\mathrm{z}}_1{\mathrm{z}}_2=100$$$$\mathrm{as}\mathrm{both}{\mathrm{z}}_1\mathrm{and}{\mathrm{z}}_2\mathrm{are}\mathrm{positive}$$$$\mathrm{we}\mathrm{have}{\mathrm{z}}_1+{\mathrm{z}}_2=10$$

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