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Question Number 181841 by mr W last updated on 01/Dec/22

$$whatislarger,\sqrt{11}+\sqrt{13}or7?$$

Answered by hmr last updated on 01/Dec/22

$$$$$$assumethat:$$$$\sqrt{11}+\sqrt{13}<7$$$$\iff$$$$11+2\sqrt{143}+13<49$$$$\iff$$$$24+2\sqrt{143}<49$$$$\iff$$$$2\sqrt{143}<25$$$$\iff$$$$\sqrt{143}<12.5$$$$andthisisalwaystrue$$$$because\sqrt{143}<\sqrt{144}=12$$$$and12<12.5$$$$so\sqrt{143}<12.5$$$$anditisequalto\sqrt{11}+\sqrt{13}<7$$

Commented by mr W last updated on 01/Dec/22

$$thanks!$$

Commented by hmr last updated on 01/Dec/22

$${you}^{\prime }rewelcome$$

Answered by BaliramKumar last updated on 01/Dec/22

$$\sqrt{11}+\sqrt{13}\stackrel{?}{=}7$$$${(\sqrt{11}+\sqrt{13})}^2\stackrel{?}{=}{7}^2$$$$11+13+2\sqrt{11\times 13}\stackrel{?}{=}49$$$$24+2\sqrt{143}\stackrel{?}{=}24+25$$$$2\sqrt{143}\stackrel{?}{=}25$$$$\sqrt{143}\stackrel{?}{=}12.5$$$$\sqrt{143}<12.5$$$$\sqrt{11}+\sqrt{13}<7$$

Commented by mr W last updated on 01/Dec/22

$$thanks!$$

Answered by mr W last updated on 01/Dec/22

$${(\sqrt{11}+\sqrt{13})}^2$$$$=11+13+2\sqrt{11\times 13}$$$$=24+2\sqrt{(12-1)(12+1)}$$$$=24+2\sqrt{{12}^2-1}$$$$<24+2\sqrt{{12}^2}$$$$<24+2\sqrt{12.5^2}$$$$=24+2\times 12.5$$$$=49$$$$=7^2$$$$\Rightarrow \sqrt{11}+\sqrt{13}<7$$

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