Which of the following rational numbers is not between 3–√ and 6–√
?
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*}
To determine which rational number is not between 3–√ and 6–√, we need to find the decimal representation of the numbers and compare them.
First, let's simplify the expressions: 3–√ = 3 – √1 = 3 – 1 = 2
And, 6–√ = 6 – √1 = 6 – 1 = 5
So, we need to find a rational number between 2 and 5.
Let's consider the given rational numbers:
a) 2.5
b) 3.2
c) 4
d) 4.5
e) 5.5
Now, let's check which of these numbers falls between 2 and 5.
a) 2.5 is between 2 and 5.
b) 3.2 is between 2 and 5.
c) 4 is between 2 and 5.
d) 4.5 is between 2 and 5.
e) 5.5 is NOT between 2 and 5.
Therefore, the rational number that is NOT between 3–√ and 6–√ is 5.5.