Let \(x=\sqrt6+\sqrt7\). What two consecutive integers does x lie between?
Figure out the answer without using a calculator. Explain in words how you arrived at your answer, and how you know for certain that it's correct. (Imagine that you are trying to convince a very skeptical person of this; a guess won't cut it.)
+128460 Note that
2.4^2 = 5.76 → √ 5.76 = 2.4
2.5^2 = 6.25 → √6.25 = 2.5
2.6^2 = 6.76 → √6.76 = 2.6
2.7^2 = 7.29 → √7.29 = 2.7
So
√ 5.76 + √6.76 = 2.4 + 2.6 = 5.0 (1)
And
√6.25 + √7.29 = 2.5 + 2.7 = 5.2 (2)
But
(1) < √6 + √7 < (2)
So √6 + √7 must lie between 5.0 and 5.2
Thus.....√6 + √7 must lie between 5 and 6
