what-is-larger-11-13-or-7-

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

w h a t i s l a r g e r, \sqrt{11} + \sqrt{13} o r 7 ?

Answered by hmr last updated on 01/Dec/22

a s s u m e t h a t :

\sqrt{11} + \sqrt{13} < 7

\Leftrightarrow

11 + 2 \sqrt{143} + 13 < 49

\Leftrightarrow

24 + 2 \sqrt{143} < 49

\Leftrightarrow

2 \sqrt{143} < 25

\Leftrightarrow

\sqrt{143} < 12.5

a n d t h i s i s a l w a y s t r u e

b e c a u s e \sqrt{143} < \sqrt{144} = 12

a n d 12 < 12.5

s o \sqrt{143} < 12.5

a n d i t i s e q u a l t o \sqrt{11} + \sqrt{13} < 7

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

t h a n k s!

Commented by hmr last updated on 01/Dec/22

{y o u}^{'} r e w e l c o m e

Answered by BaliramKumar last updated on 01/Dec/22

\sqrt{11} + \sqrt{13} \overset{?}{=} 7

{(\sqrt{11} + \sqrt{13})}^{2} \overset{?}{=} 7^{2}

11 + 13 + 2 \sqrt{11 \times 13} \overset{?}{=} 49

24 + 2 \sqrt{143} \overset{?}{=} 24 + 25

2 \sqrt{143} \overset{?}{=} 25

\sqrt{143} \overset{?}{=} 12.5

\sqrt{143} < 12.5

\sqrt{11} + \sqrt{13} < 7

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

t h a n k s!

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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