Show an example of a rational number that is between 10−−√ and 11−−√.
To find a rational number between $\sqrt{10}$ and $\sqrt{11}$, we can take the average of the two numbers. Since $\sqrt{10} < 3 < \sqrt{11}$, we know that $3$ is between $\sqrt{10}$ and $\sqrt{11}$. So, we can take $\frac{\sqrt{10} + \sqrt{11}}{2}$ as our example. We can rationalize the denominator by multiplying the numerator and denominator by $\sqrt{11} - \sqrt{10}$: \begin{align*}
\frac{\sqrt{10} + \sqrt{11}}{2} \cdot \frac{\sqrt{11} - \sqrt{10}}{\sqrt{11} - \sqrt{10}} &= \frac{\sqrt{10}\sqrt{11} - \sqrt{11}\sqrt{10} + \sqrt{10}\sqrt{11} - \sqrt{10}\sqrt{10}}{2(\sqrt{11} - \sqrt{10})} \\
&= \frac{2\sqrt{10}\sqrt{11}-10}{2(\sqrt{11} - \sqrt{10})} \\
&= \frac{\sqrt{10}\sqrt{11}-5}{\sqrt{11} - \sqrt{10}},
\end{align*} where $2\sqrt{10}\sqrt{11}-10$ is rational. Since $\sqrt{11}-\sqrt{10}$ is irrational, we can multiply both the numerator and denominator by its conjugate to rationalize the denominator further: \begin{align*}
\frac{\sqrt{10}\sqrt{11}-5}{\sqrt{11} - \sqrt{10}}\cdot\frac{\sqrt{11}+\sqrt{10}}{\sqrt{11}+\sqrt{10}} &= \frac{(\sqrt{10})(\sqrt{11})(\sqrt{11}+\sqrt{10})-5(\sqrt{11}+\sqrt{10})}{(\sqrt{11} - \sqrt{10})(\sqrt{11}+\sqrt{10})} \\
&= \boxed{\frac{5\sqrt{10}+11}{1}}.
\end{align*}
Show an example of a rational number that is between 10−−√ and 11−−√.
To find a rational number between $\sqrt{10}$ and $\sqrt{11}$, we can use the property that the square root of any positive integer is always irrational, unless the integer is a perfect square. Both $10$ and $11$ are not perfect squares, so their square roots are irrational numbers.
However, we can approximate $\sqrt{10}$ and $\sqrt{11}$ using decimals to find a rational number between them. We know that $\sqrt{10} \approx 3.16$ and $\sqrt{11} \approx 3.32$. Taking the average of these two approximations, we get $\frac{3.16 + 3.32}{2} = 3.24$. This is our desired rational number between $\sqrt{10}$ and $\sqrt{11}$.
To find a rational number between √10 and √11, we can take the average of these two square roots.
Average = (√10 + √11) / 2
Using a calculator, we can calculate the exact value of this average:
Average ≈ (3.162 + 3.317) / 2 ≈ 6.4795
So, the rational number that is between √10 and √11 is approximately 6.4795.