Using a number line, what whole number and rational number to the nearest half is √105 between?
To determine the whole number and rational number to the nearest half that √105 is between, we first need to find the closest whole number and rational number to √105.
To find the whole number, we can estimate the square root of 105. The closest whole number would be the square root of 100, which is 10.
To find the rational number to the nearest half, we need to determine if √105 is closer to 10 or 11. We can square both numbers to compare:
10 squared = 100
11 squared = 121
Since 105 is closer to 100 than 121, √105 is closer to 10.
Therefore, √105 is between the whole number 10 and the rational number 10.5 on a number line.
Using rational approximations, what statement is true?
A. √48 > √36
B. √49 > 7
C. √49 < 7
D. √48 < √36
To compare the values using rational approximations, we can estimate the square roots of 48 and 36.
For √48:
We know that 6 squared is 36, and 7 squared is 49, so the square root of 48 must be between 6 and 7.
We can approximate it as √48 ≈ 6.93.
For √36:
Since 6 squared is 36, the square root of 36 is exactly 6.
Comparing the approximations, we find that √48 ≈ 6.93 is greater than √36 = 6.
Therefore, the correct statement is:
A. √48 > √36
Using rational approximations, what statement is true?
A. √71 > √61
B. √71 < √61
C. √81 > 9
D. √81 < 9
To compare the values using rational approximations, we can estimate the square roots of 71, 61, and 81.
For √71:
We know that 8 squared is 64 and 9 squared is 81, so the square root of 71 must be between 8 and 9.
We can approximate it as √71 ≈ 8.44.
For √61:
We know that 7 squared is 49 and 8 squared is 64, so the square root of 61 must be between 7 and 8.
We can approximate it as √61 ≈ 7.81.
For √81:
The square root of 81 is exactly 9.
Comparing the approximations, we find that √71 ≈ 8.44 is greater than √61 ≈ 7.81.
Therefore, the correct statement is:
A. √71 > √61
Using rational approximations, what statement is true?
A. √16 > 4
B. √12 > pi
C. √16 < 4
D. √12 < pi
To compare the values using rational approximations, we can estimate the square roots of 16 and 12, and compare them to their known values.
For √16:
The square root of 16 is exactly 4.
For √12:
We know that 3 squared is 9 and 4 squared is 16, so the square root of 12 must be between 3 and 4.
We can approximate it as √12 ≈ 3.46.
Comparing the values, we find that √16 = 4 is equal to 4, and √12 ≈ 3.46 is less than 4.
Therefore, the correct statement is:
C. √16 < 4
the last one is 12 −−−√ >π
To find the whole number and rational number to the nearest half that is between √105, we can start by finding the square root of 105.
Using a calculator or a math software, we can calculate that the square root of 105 is approximately 10.2469507659.
Next, let's think about the number line and identify the whole numbers and rational numbers to the nearest half that are on either side of √105.
The whole numbers and rational numbers closest to 10.2469507659 to the left and right are:
To the left: 10 and 10.5
To the right: 11 and 10.5
Therefore, the whole number and rational numbers to the nearest half that √105 is between are 10 and 10.5.