From the shape of the graph f(x)=√x one can see the function increases but at a decreasing rate.
So √6 is bigger than √5
and √8 is bigger than √7 but less so.
√14 is bigger than √12 but less than twice as much as either of the above.
So the second expression is greater.
In fact, you can replace the 12 and 14 by numbers as small as 6 and 8 and the second expression would still be bigger:
√5 + √7  √6 and √6+ √8  √8
because √7  √6 < √6  √5.

Posted by Jer
on 20150105 09:02:01 