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.)

Guest Jul 7, 2017

#1**+1 **

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

CPhill Jul 7, 2017