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# Help needed

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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
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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

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