Question

PLEASE HELP!!!!!DUE TOMORROW
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.)

Answer 1

The two consecutive integers that x lies between is 5 and 6

Explanation:

The given expression is $x=\sqrt{6} +\sqrt{7}$

Since, the perfect squares of $2.4, 2.5, 2.6$ and $2.7$ are given by

$2.4^2=5.76$ ⇒ $\sqrt{5.76} =2.4$

$2.5^2=6.25$ ⇒ $\sqrt{6.25} =2.5$

$2.6^2=6.76$ ⇒ $\sqrt{6.76} =2.6$

$2.7^2=7.29$ ⇒ $\sqrt{7.29} =2.7$

From the above perfect squares, the values close to $\sqrt{6}$ and $\sqrt{7}$ are given by

$x=\sqrt{5.76} +\sqrt{6.76}\\x =2.4+2.6\\x=5$ ------------(1)

Since, the expression is $x=\sqrt{6} +\sqrt{7}$

Substituting the values, we have,

$x=2.45+2.65\\x=5.10$        -------------(2)

Thus, from (1) and (2), the values of x lies between 5 and 6

Hence, the two consecutive integers that x lies between is 5 and 6