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

what is larger, (√(11))+(√(13)) or 7?

$${what}\:{is}\:{larger},\:\sqrt{\mathrm{11}}+\sqrt{\mathrm{13}}\:{or}\:\mathrm{7}? \\ $$

Answered by hmr last updated on 01/Dec/22

assume that: (√(11)) + (√(13 )) < 7 ⇔ 11 + 2(√(143)) + 13 < 49 ⇔ 24 + 2(√(143)) < 49 ⇔ 2(√(143)) < 25 ⇔ (√(143)) < 12.5 and this is always true because (√(143)) < (√(144)) = 12 and 12 < 12.5 so (√(143)) < 12.5 and it is equal to (√(11 ))+ (√(13)) < 7

$$ \\ $$$${assume}\:{that}:\: \\ $$$$\sqrt{\mathrm{11}}\:+\:\sqrt{\mathrm{13}\:}\:<\:\mathrm{7} \\ $$$$\Leftrightarrow \\ $$$$\mathrm{11}\:+\:\mathrm{2}\sqrt{\mathrm{143}}\:+\:\mathrm{13}\:<\:\mathrm{49} \\ $$$$\Leftrightarrow \\ $$$$\mathrm{24}\:+\:\mathrm{2}\sqrt{\mathrm{143}}\:<\:\mathrm{49} \\ $$$$\Leftrightarrow \\ $$$$\mathrm{2}\sqrt{\mathrm{143}}\:<\:\mathrm{25} \\ $$$$\Leftrightarrow \\ $$$$\sqrt{\mathrm{143}}\:<\:\mathrm{12}.\mathrm{5} \\ $$$${and}\:{this}\:{is}\:{always}\:{true} \\ $$$${because}\:\sqrt{\mathrm{143}}\:<\:\sqrt{\mathrm{144}}\:=\:\mathrm{12} \\ $$$${and}\:\mathrm{12}\:<\:\mathrm{12}.\mathrm{5} \\ $$$${so}\:\sqrt{\mathrm{143}}\:<\:\mathrm{12}.\mathrm{5} \\ $$$${and}\:{it}\:{is}\:{equal}\:{to}\:\sqrt{\mathrm{11}\:}+\:\sqrt{\mathrm{13}}\:<\:\mathrm{7} \\ $$

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

thanks!

$${thanks}! \\ $$

Commented by hmr last updated on 01/Dec/22

you′re welcome

$${you}'{re}\:{welcome}\: \\ $$

Answered by BaliramKumar last updated on 01/Dec/22

(√(11))+(√(13)) =^? 7 ((√(11)) + (√(13)))^2 =^? 7^2 11+13+2(√(11×13)) =^? 49 24 +2(√(143)) =^? 24+25 2(√(143)) =^? 25 (√(143)) =^? 12.5 (√(143)) < 12.5 (√(11)) + (√(13)) < 7

$$\sqrt{\mathrm{11}}+\sqrt{\mathrm{13}}\:\overset{?} {=}\:\mathrm{7} \\ $$$$\left(\sqrt{\mathrm{11}}\:+\:\sqrt{\mathrm{13}}\right)^{\mathrm{2}} \:\overset{?} {=}\:\mathrm{7}^{\mathrm{2}} \\ $$$$\mathrm{11}+\mathrm{13}+\mathrm{2}\sqrt{\mathrm{11}×\mathrm{13}}\:\overset{?} {=}\:\mathrm{49} \\ $$$$\mathrm{24}\:+\mathrm{2}\sqrt{\mathrm{143}}\:\overset{?} {=}\:\mathrm{24}+\mathrm{25} \\ $$$$\mathrm{2}\sqrt{\mathrm{143}}\:\overset{?} {=}\:\:\mathrm{25} \\ $$$$\sqrt{\mathrm{143}}\:\overset{?} {=}\:\mathrm{12}.\mathrm{5} \\ $$$$\sqrt{\mathrm{143}}\:<\:\mathrm{12}.\mathrm{5} \\ $$$$\sqrt{\mathrm{11}}\:+\:\sqrt{\mathrm{13}}\:<\:\mathrm{7} \\ $$

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

thanks!

$${thanks}! \\ $$

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

((√(11))+(√(13)))^2 =11+13+2(√(11×13)) =24+2(√((12−1)(12+1))) =24+2(√(12^2 −1)) <24+2(√(12^2 )) <24+2(√(12.5^2 )) =24+2×12.5 =49 =7^2 ⇒(√(11))+(√(13))<7

$$\left(\sqrt{\mathrm{11}}+\sqrt{\mathrm{13}}\right)^{\mathrm{2}} \\ $$$$=\mathrm{11}+\mathrm{13}+\mathrm{2}\sqrt{\mathrm{11}×\mathrm{13}} \\ $$$$=\mathrm{24}+\mathrm{2}\sqrt{\left(\mathrm{12}−\mathrm{1}\right)\left(\mathrm{12}+\mathrm{1}\right)} \\ $$$$=\mathrm{24}+\mathrm{2}\sqrt{\mathrm{12}^{\mathrm{2}} −\mathrm{1}} \\ $$$$<\mathrm{24}+\mathrm{2}\sqrt{\mathrm{12}^{\mathrm{2}} } \\ $$$$<\mathrm{24}+\mathrm{2}\sqrt{\mathrm{12}.\mathrm{5}^{\mathrm{2}} } \\ $$$$=\mathrm{24}+\mathrm{2}×\mathrm{12}.\mathrm{5} \\ $$$$=\mathrm{49} \\ $$$$=\mathrm{7}^{\mathrm{2}} \\ $$$$\Rightarrow\sqrt{\mathrm{11}}+\sqrt{\mathrm{13}}<\mathrm{7} \\ $$

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