Which of the following rational numbers is not between 3–√ and 6–√

Question

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Answer 1

To determine which rational number is not between $3-\sqrt{2} = \sqrt{2}+1$ and $6-\sqrt{2}$, we first rationalize the denominators of the numbers $3-\sqrt{2}$ and $6-\sqrt{2}$. \begin{align*}
3-\sqrt{2} &= \left(3-\sqrt{2}\right) \cdot \frac{3+\sqrt{2}}{3+\sqrt{2}} \\
&= \frac{9+\sqrt{18}-3\sqrt{2}-2}{1} \\
&= \frac{7+\sqrt{18}-3\sqrt{2}}{1} \\
&= \frac{7-3\sqrt{2}+\sqrt{9\cdot 2}}{1} \\
&= \frac{7-3\sqrt{2}+\sqrt{9}\cdot \sqrt{2}}{1} \\
&= \frac{7-3\sqrt{2}+3\sqrt{2}}{1} \\
&= \boxed{7}.
\end{align*}