Find a rational number with one decimal place that is between 17−−√ and 18−−√
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To find a rational number between $\sqrt{17}$ and $\sqrt{18}$, we can "rationalize" the denominator. To do this, we multiply the numerator and denominator of $\frac{1}{\sqrt{17}}$ by $\sqrt{17}$ to get \begin{align*}
\frac{1}{\sqrt{17}} &= \frac{1}{\sqrt{17}}\cdot \frac{\sqrt{17}}{\sqrt{17}}\\
&= \frac{\sqrt{17}}{17}.
\end{align*} Similarly, to rationalize the denominator of $\frac{1}{\sqrt{18}}$, we multiply the numerator and denominator of $\frac{1}{\sqrt{18}}$ by $\sqrt{18}$ to get \begin{align*}
\frac{1}{\sqrt{18}} &= \frac{1}{\sqrt{18}}\cdot \frac{\sqrt{18}}{\sqrt{18}}\\
&= \frac{\sqrt{18}}{18}\\
&= \frac{3\sqrt{2}}{18}\\
&= \frac{\sqrt{2}}{6}.
\end{align*} Therefore, the rational number between $\sqrt{17}$ and $\sqrt{18}$ is $\boxed{\dfrac{\sqrt{17}}{17}}$.
Find a rational number with one decimal place that is between 17−−√ and 18−−√
To find a rational number between $\sqrt{17}$ and $\sqrt{18}$, we can find the average of these two numbers. Since $\sqrt{17} \approx 4.1$ and $\sqrt{18} \approx 4.2,$ the average is $\frac{4.1+4.2}{2} = 4.15,$ which is a decimal number with one decimal place. Therefore, a rational number with one decimal place that is between $\sqrt{17}$ and $\sqrt{18}$ is $\boxed{4.2}.$
Which of the following is between π and 11−−√?(1 point)
To determine which number is between $\pi$ and $\sqrt{11},$ we can find their average. Since $\pi \approx 3.14$ and $\sqrt{11} \approx 3.32,$ the average is $\frac{3.14+3.32}{2} = 3.23.$ Therefore, the number $3.23$ is between $\pi$ and $\sqrt{11}.$ Thus, the answer is $\boxed{3.23}.$
Which of the following is between π and 11−−√?(1 point)
10−−√
12−−√
7–√
8–√
To determine which number is between $\pi$ and $\sqrt{11}$, we can compare their values. We have $\pi \approx 3.14$ and $\sqrt{11} \approx 3.32$. Therefore, the number $10-\sqrt{2}$ is not between $\pi$ and $\sqrt{11}$. The number $12-\sqrt{2}$ is greater than $\sqrt{11}$, so it is also not between $\pi$ and $\sqrt{11}$. The number $7-\sqrt{2}$ is less than $\pi$, so it is also not between $\pi$ and $\sqrt{11}$. However, we can see that $8-\sqrt{2}$, which is approximately $8-1.41=6.68$, is between $\pi$ and $\sqrt{11}$. Therefore, the answer is $\boxed{8-\sqrt{2}}$.
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Show an example of an irrational number between 9 and 11.(1 point)
A well-known irrational number is $\pi \approx 3.14159$.
Since $9 < \pi < 11$, we can see that $\pi$ is an example of an irrational number between 9 and 11.
Show an example of an irrational number between 9 and 11.(1 point)
100−−−√
10−−√
99−−√
20−−√
To find an example of an irrational number between 9 and 11, we can look at the square root of a number.
The only option that represents an irrational number is $\boxed{10-\sqrt{2}}$.
What is the least integer whose square root is an irrational number between 5 and 7?(1 point)
Responses
26
13
25
6
To find the least integer whose square root is an irrational number between 5 and 7, we need to find the smallest perfect square greater than 5 and less than 7. The perfect square immediately greater than 5 is 6.25, which corresponds to the least integer whose square root is an irrational number between 5 and 7. Since $6.25 = 2.5^2$, the least integer is $\boxed{25}$.
Show an example of a rational number that is between 10−−√ and 11−−√.(1 point)
10/3
16/5
3.1
3.4
To find a rational number between $\sqrt{10}$ and $\sqrt{11}$, we can take the average of these two numbers. Since $\sqrt{10} \approx 3.16$ and $\sqrt{11} \approx 3.32$, the average is $\frac{3.16 + 3.32}{2} = 3.24$.
Out of the options provided, the only number that is approximately equal to 3.24 is $\boxed{3.1}$.
Which irrational number is between 14−−√ and 18−−√?(1 point)
15−−√
13−−√
10−−√
19−−√
To find an irrational number between $\sqrt{14}$ and $\sqrt{18}$, we can look for an irrational number whose square falls within this range. We know that $\left(\sqrt{15}\right)^2 \approx 15$, which is between 14 and 18. Therefore, an irrational number between $\sqrt{14}$ and $\sqrt{18}$ is $\boxed{\sqrt{15}}$.
How many irrational numbers are there between 11−−√ and 17−−√?(1 point)
twenty-eight numbers
six numbers
four numbers
an infinite number