Question
Using a number line, what whole number and rational number to the nearest half is 97−−√ between?
Answers
Answer
Using a number line, what whole number and rational number to the nearest half is 97−−√ between?
Answer
Using rational approximations, what is an inequality statement comparing 21−−√ and 17−−√? Enter <, >, or =.
Answer
Using a number line, what is the approximate value of 19−−√?(1 point)
Answer
Using a number line, what is the approximate value of 19−−√?(1 point)
Responses
4.13
4.5
4.38
4.25
Responses
4.13
4.5
4.38
4.25
Answer
Using a number line, what whole number and rational number to the nearest half is 105−−−√ between?(1 point)
Answer
Using rational approximations, what statement is true?(1 point)
Responses
48−−√<36−−√
start root 48 end root less than start root 36 end root
49−−√>7
start root 49 end root greater than 7
49−−√<7
start root 49 end root less than 7
48−−√>36−−√
Responses
48−−√<36−−√
start root 48 end root less than start root 36 end root
49−−√>7
start root 49 end root greater than 7
49−−√<7
start root 49 end root less than 7
48−−√>36−−√
Answer
Using rational approximations, what statement is true?(1 point)
Responses
81−−√>9
start root 81 end root greater than 9
81−−√<9
start root 81 end root less than 9
71 −−−√<61−−√
start root 71 end root less than start root 61 end root
71−−√>61−−√
Responses
81−−√>9
start root 81 end root greater than 9
81−−√<9
start root 81 end root less than 9
71 −−−√<61−−√
start root 71 end root less than start root 61 end root
71−−√>61−−√
Answer
i think this is wrong
Answer
Using rational approximations, what statement is true?(1 point)
Responses
16 −−−√<4
start root 16 end root less than 4
12 −−−√ >π
start root 12 end root greater than pi
16−−√>4
start root 16 end root greater than 4
12 −−−√<π
Responses
16 −−−√<4
start root 16 end root less than 4
12 −−−√ >π
start root 12 end root greater than pi
16−−√>4
start root 16 end root greater than 4
12 −−−√<π
Answer
Using a number line, what whole number and rational number to the nearest half is 97−−√ between?
Answer
bot
Answered by
GPT 3.5
To find the whole number and rational number to the nearest half that is between $\sqrt{97}$, we can look for the two integers between which $\sqrt{97}$ lies.
First, we note that $10^2=100$ is larger than $97$. Since $\sqrt{100}=10$, $\sqrt{97}$ is a little less than 10.
We can compute by hand or with a calculator that $9^2=81$ and $10^2=100$, so $\sqrt{81}<\sqrt{97}<\sqrt{100}$.
Therefore, the whole number that $\sqrt{97}$ is between is 9.
Next, we can estimate the fractional part by dividing up the interval between 9 and 10 into halves. Since $\sqrt{97}$ is closer to 10, the fractional part will be closer to one than to zero. Therefore, $\sqrt{97}$ is closer to $9\dfrac{1}{2}$ than to $9\dfrac{0}{2}$.
So, the whole number and rational number to the nearest half that $\sqrt{97}$ is between are $\boxed{9\text{ and }\dfrac{9}{2}}$.
First, we note that $10^2=100$ is larger than $97$. Since $\sqrt{100}=10$, $\sqrt{97}$ is a little less than 10.
We can compute by hand or with a calculator that $9^2=81$ and $10^2=100$, so $\sqrt{81}<\sqrt{97}<\sqrt{100}$.
Therefore, the whole number that $\sqrt{97}$ is between is 9.
Next, we can estimate the fractional part by dividing up the interval between 9 and 10 into halves. Since $\sqrt{97}$ is closer to 10, the fractional part will be closer to one than to zero. Therefore, $\sqrt{97}$ is closer to $9\dfrac{1}{2}$ than to $9\dfrac{0}{2}$.
So, the whole number and rational number to the nearest half that $\sqrt{97}$ is between are $\boxed{9\text{ and }\dfrac{9}{2}}$.
Answered by
GPT 3.5
To find the whole number and rational number to the nearest half that $\sqrt{97}$ is between, we can use a number line.
First, we know that $9^2 = 81$ and $10^2 = 100$. Therefore, $\sqrt{97}$ must be between 9 and 10.
Next, we can divide the interval between 9 and 10 into halves. The midpoint of this interval is $9\frac{1}{2}$, which is exactly halfway between 9 and 10.
Since $\sqrt{97}$ is closer to 10 than to 9, the nearest whole number is 10.
Looking at the halves, we see that $\sqrt{97}$ is closer to $9\frac{1}{2}$ than to $9\frac{0}{2}$.
Therefore, the whole number and rational number to the nearest half that $\sqrt{97}$ is between are $\boxed{10\text{ and } 9\frac{1}{2}}$.
First, we know that $9^2 = 81$ and $10^2 = 100$. Therefore, $\sqrt{97}$ must be between 9 and 10.
Next, we can divide the interval between 9 and 10 into halves. The midpoint of this interval is $9\frac{1}{2}$, which is exactly halfway between 9 and 10.
Since $\sqrt{97}$ is closer to 10 than to 9, the nearest whole number is 10.
Looking at the halves, we see that $\sqrt{97}$ is closer to $9\frac{1}{2}$ than to $9\frac{0}{2}$.
Therefore, the whole number and rational number to the nearest half that $\sqrt{97}$ is between are $\boxed{10\text{ and } 9\frac{1}{2}}$.
Answered by
GPT 3.5
To compare $\sqrt{21}$ and $\sqrt{17}$, we can use rational approximations.
For $\sqrt{21}$, we can estimate that $21$ is between $16$ and $25$, and $\sqrt{16}=4$ and $\sqrt{25}=5$. Therefore, $4< \sqrt{21}< 5$.
For $\sqrt{17}$, we can estimate that $17$ is between $16$ and $18$, and $\sqrt{16}=4$ and $\sqrt{18}\approx4.24$. Therefore, $4< \sqrt{17}<4.24$.
Comparing the two values, we have $4< \sqrt{17}< \sqrt{21}<5$.
Thus, the inequality statement comparing $\sqrt{21}$ and $\sqrt{17}$ is $\sqrt{17}<\sqrt{21}$.
For $\sqrt{21}$, we can estimate that $21$ is between $16$ and $25$, and $\sqrt{16}=4$ and $\sqrt{25}=5$. Therefore, $4< \sqrt{21}< 5$.
For $\sqrt{17}$, we can estimate that $17$ is between $16$ and $18$, and $\sqrt{16}=4$ and $\sqrt{18}\approx4.24$. Therefore, $4< \sqrt{17}<4.24$.
Comparing the two values, we have $4< \sqrt{17}< \sqrt{21}<5$.
Thus, the inequality statement comparing $\sqrt{21}$ and $\sqrt{17}$ is $\sqrt{17}<\sqrt{21}$.
Answered by
GPT 3.5
To approximate the value of $\sqrt{19}$ using a number line, we can start by identifying the two perfect square numbers that $\sqrt{19}$ falls between.
Since $4^2=16$ and $5^2=25$, we know that $\sqrt{19}$ is between 4 and 5.
To get a more precise estimate, we can divide the interval between 4 and 5 into smaller parts. One way to do this is by halves. So, we can divide the interval into 2 equal parts:
$$4 \quad \frac{4+5}{2} \quad 5.$$
Calculating the average of 4 and 5, we get $\frac{4+5}{2} = \frac{9}{2} = 4.5$.
Therefore, the approximate value of $\sqrt{19}$ is $\boxed{4.5}$.
Since $4^2=16$ and $5^2=25$, we know that $\sqrt{19}$ is between 4 and 5.
To get a more precise estimate, we can divide the interval between 4 and 5 into smaller parts. One way to do this is by halves. So, we can divide the interval into 2 equal parts:
$$4 \quad \frac{4+5}{2} \quad 5.$$
Calculating the average of 4 and 5, we get $\frac{4+5}{2} = \frac{9}{2} = 4.5$.
Therefore, the approximate value of $\sqrt{19}$ is $\boxed{4.5}$.
Answered by
GPT 3.5
To approximate the value of $\sqrt{19}$ using a number line, we can start by identifying the two perfect square numbers that $\sqrt{19}$ falls between.
Since $4^2=16$ and $5^2=25$, we know that $\sqrt{19}$ is between 4 and 5.
To get a more precise estimate, we can divide the interval between 4 and 5 into smaller parts. One reasonable choice is to divide the interval into tenths:
$$4.0 \quad 4.1 \quad 4.2 \quad 4.3 \quad 4.4 \quad 4.5 \quad 4.6 \quad 4.7 \quad 4.8 \quad 4.9 \quad 5.0.$$
From this division, we see that $\sqrt{19}$ is closer to 4.3 than 4.2, so it is reasonable to estimate the value as approximately $\boxed{4.3}$.
Since $4^2=16$ and $5^2=25$, we know that $\sqrt{19}$ is between 4 and 5.
To get a more precise estimate, we can divide the interval between 4 and 5 into smaller parts. One reasonable choice is to divide the interval into tenths:
$$4.0 \quad 4.1 \quad 4.2 \quad 4.3 \quad 4.4 \quad 4.5 \quad 4.6 \quad 4.7 \quad 4.8 \quad 4.9 \quad 5.0.$$
From this division, we see that $\sqrt{19}$ is closer to 4.3 than 4.2, so it is reasonable to estimate the value as approximately $\boxed{4.3}$.
Answered by
GPT 3.5
To find the whole number and rational number to the nearest half that $\sqrt{105}$ is between, we can use a number line.
First, we note that $10^2=100$ is smaller than $105$. Since $\sqrt{100}=10$, $\sqrt{105}$ is a little greater than 10.
Next, we can estimate the whole number by rounding the square root to the nearest whole number. Since $\sqrt{105}$ is closer to 10 than to 11, the whole number that $\sqrt{105}$ is between is 10.
Now, we divide the interval between 10 and 11 into halves to find the rational number to the nearest half. The midpoint of this interval is $10\frac{1}{2}$, which is exactly halfway between 10 and 11.
Since $\sqrt{105}$ is closer to 10 than to 11, the nearest whole number is 10.
Looking at the halves, we see that $\sqrt{105}$ is closer to $10\frac{1}{2}$ than to $10\frac{0}{2}$.
Therefore, the whole number and rational number to the nearest half that $\sqrt{105}$ is between are $\boxed{10\text{ and }10\frac{1}{2}}$.
First, we note that $10^2=100$ is smaller than $105$. Since $\sqrt{100}=10$, $\sqrt{105}$ is a little greater than 10.
Next, we can estimate the whole number by rounding the square root to the nearest whole number. Since $\sqrt{105}$ is closer to 10 than to 11, the whole number that $\sqrt{105}$ is between is 10.
Now, we divide the interval between 10 and 11 into halves to find the rational number to the nearest half. The midpoint of this interval is $10\frac{1}{2}$, which is exactly halfway between 10 and 11.
Since $\sqrt{105}$ is closer to 10 than to 11, the nearest whole number is 10.
Looking at the halves, we see that $\sqrt{105}$ is closer to $10\frac{1}{2}$ than to $10\frac{0}{2}$.
Therefore, the whole number and rational number to the nearest half that $\sqrt{105}$ is between are $\boxed{10\text{ and }10\frac{1}{2}}$.
Answered by
GPT 3.5
To compare $\sqrt{48}$ and $\sqrt{36}$ using rational approximations:
Since $7^2 = 49$ and $6^2 = 36$, we know that $\sqrt{48}$ is between $\sqrt{36}$ and $\sqrt{49}$.
Considering rational approximations, $\sqrt{36} = 6$ and $\sqrt{49} = 7$. Therefore, $\sqrt{36} < \sqrt{48} < \sqrt{49}$.
So, the correct statement is $\boxed{\sqrt{48} > \sqrt{36}}$.
Since $7^2 = 49$ and $6^2 = 36$, we know that $\sqrt{48}$ is between $\sqrt{36}$ and $\sqrt{49}$.
Considering rational approximations, $\sqrt{36} = 6$ and $\sqrt{49} = 7$. Therefore, $\sqrt{36} < \sqrt{48} < \sqrt{49}$.
So, the correct statement is $\boxed{\sqrt{48} > \sqrt{36}}$.
Answered by
GPT 3.5
To compare $\sqrt{81}$ and $\sqrt{9}$ using rational approximations:
Since $9^2 = 81$ and $\sqrt{9} = 9$, we know that $\sqrt{81}$ is equal to $9$.
Therefore, the correct statement is $\boxed{\sqrt{81} = 9}$.
Since $9^2 = 81$ and $\sqrt{9} = 9$, we know that $\sqrt{81}$ is equal to $9$.
Therefore, the correct statement is $\boxed{\sqrt{81} = 9}$.
Answered by
GPT 3.5
I apologize for the mistake. Let's correct it.
To compare $\sqrt{81}$ and $\sqrt{9}$:
$81$ is a perfect square, so $\sqrt{81}$ is equal to $9$.
$9$ is also a perfect square, so $\sqrt{9}$ is equal to $3$.
Therefore, the correct statement is $\boxed{\sqrt{81} = 9 > \sqrt{9} = 3}$.
To compare $\sqrt{81}$ and $\sqrt{9}$:
$81$ is a perfect square, so $\sqrt{81}$ is equal to $9$.
$9$ is also a perfect square, so $\sqrt{9}$ is equal to $3$.
Therefore, the correct statement is $\boxed{\sqrt{81} = 9 > \sqrt{9} = 3}$.
Answered by
GPT 3.5
To compare $\sqrt{16}$ and $\pi$ using rational approximations:
Since $4^2 = 16$, we know that $\sqrt{16}$ is equal to $4$.
Approximating the value of $\pi$, we have $\pi \approx 3.14$.
Therefore, the correct statement is $\boxed{\sqrt{16} = 4 > \pi \approx 3.14}$.
Since $4^2 = 16$, we know that $\sqrt{16}$ is equal to $4$.
Approximating the value of $\pi$, we have $\pi \approx 3.14$.
Therefore, the correct statement is $\boxed{\sqrt{16} = 4 > \pi \approx 3.14}$.
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