What is the largest integer whose square root is an irrational number between 3 and 4?(1 point)
The largest integer whose square root is an irrational number between 3 and 4 is 15.
Find a rational number between 2 and 214 using the method of adding the numbers together and dividing by 2. Write the answer as a mixed number.
Find a rational number with one decimal place that is between 17√ and 18−−√.
To find a rational number with one decimal place that is between $\sqrt{17}$ and $\sqrt{18}$, we can take the average of the two irrational numbers.
Average of $\sqrt{17}$ and $\sqrt{18}$:
$$\frac{\sqrt{17}+\sqrt{18}}{2}$$
We can simplify the expression by multiplying the numerator and denominator by the conjugate of the numerator:
$$\frac{\sqrt{17}+\sqrt{18}}{2} \cdot \frac{\sqrt{17}-\sqrt{18}}{\sqrt{17}-\sqrt{18}}$$
Simplifying further:
$$\frac{17-18}{2(\sqrt{17}-\sqrt{18})}$$
$$\frac{-1}{2(\sqrt{17}-\sqrt{18})}$$
We can now approximate this rational number to one decimal place.
Calculating $\sqrt{17} \approx 4.12$ and $\sqrt{18} \approx 4.24$, we can substitute these approximations into the rational number expression:
$$\frac{-1}{2(4.12-4.24)}$$
$$\frac{-1}{2(-0.12)}$$
$$\frac{-1}{-0.24}$$
Simplifying further:
$$\frac{1}{0.24}$$
Approximating this rational number to one decimal place, we get:
$$\frac{1}{0.24} \approx 4.2$$
Therefore, a rational number with one decimal place that is between $\sqrt{17}$ and $\sqrt{18}$ is 4.2.
Show an example of an irrational number between 9 and 11.(1 point)
One example of an irrational number between 9 and 11 is the square root of 10, denoted as $\sqrt{10}$.
What is the least integer whose square root is an irrational number between 5 and 7?
To find the least integer whose square root is an irrational number between 5 and 7, we need to determine the least perfect square greater than 25 (5 squared) and the greatest perfect square less than 49 (7 squared).
The least perfect square greater than 25 is 36 (6 squared), and the greatest perfect square less than 49 is 49 itself (7 squared).
Therefore, the least integer whose square root is an irrational number between 5 and 7 is 7, since the square root of 49 is an irrational number and falls within the range provided.
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 irrational numbers.
Average of $\sqrt{10}$ and $\sqrt{11}$:
$$\frac{\sqrt{10}+\sqrt{11}}{2}$$
We can simplify the expression by multiplying the numerator and denominator by the conjugate of the numerator:
$$\frac{\sqrt{10}+\sqrt{11}}{2} \cdot \frac{\sqrt{10}-\sqrt{11}}{\sqrt{10}-\sqrt{11}}$$
Simplifying further:
$$\frac{10-11}{2(\sqrt{10}-\sqrt{11})}$$
$$\frac{-1}{2(\sqrt{10}-\sqrt{11})}$$
We can approximate this rational number to obtain a decimal representation.
Calculating $\sqrt{10} \approx 3.16$ and $\sqrt{11} \approx 3.32$, we can substitute these approximations into the rational number expression:
$$\frac{-1}{2(3.16-3.32)}$$
$$\frac{-1}{2(-0.16)}$$
$$\frac{-1}{-0.32}$$
Simplifying further:
$$\frac{1}{0.32}$$
Approximating this rational number to two decimal places, we get:
$$\frac{1}{0.32} \approx 3.13$$
Therefore, a rational number between $\sqrt{10}$ and $\sqrt{11}$ is approximately 3.13.