using rational approximations, what is an inequality statement comparing √21 and √17
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Using rational approximations, we can compare the values of √21 and √17.
√21 ≈ 4.5826
√17 ≈ 4.1231
Comparing the two values, we can see that √21 is greater than √17.
√21 > √17
Thank you! Can you answer this question for me? For some reason it says that the answer isnt 10 and 9.5
Using a number line, what whole number and rational number to the nearest half is √97 between?
To find a whole number and a rational number to the nearest half that √97 falls between, let's examine the square roots of nearby perfect squares.
The perfect square just below 97 is 81, with a square root of 9.
The perfect square just above 97 is 100, with a square root of 10.
Since √97 is greater than 9 (√81) and less than 10 (√100), we can conclude that the whole number √97 falls between is 9.
To find a rational number to the nearest half, we can look at √97's decimal approximation:
√97 ≈ 9.8488
Rounding 9.8488 to the nearest half gives us 10.
Therefore, √97 falls between the whole number 9 and the rational number 10.
To compare √21 and √17 using rational approximations, we can find the square roots of 21 and 17 using a calculator or by using a series of rational approximations.
Approximating √21:
We can start with a reasonable guess, like √16 = 4. Then, refine the approximation using the Babylonian method:
(4 + 21/4) / 2 = 5.25
(5.25 + 21/5.25) / 2 = 4.91
(4.91 + 21/4.91) / 2 = 4.58
(4.58 + 21/4.58) / 2 = 4.58 (approx.)
Approximating √17:
Similarly, we can start with a guess, like √16 = 4, and refine the approximation:
(4 + 17/4) / 2 = 4.625
(4.625 + 17/4.625) / 2 = 4.123
(4.123 + 17/4.123) / 2 = 4.123 (approx.)
Comparing √21 and √17:
Based on these approximations, we can see that √21 is approximately 4.58, while √17 is approximately 4.123.
Since 4.58 is greater than 4.123, we can conclude that √21 is greater than √17.
Therefore, the inequality statement is √21 > √17.
To compare √21 and √17 using rational approximations, we can calculate their decimal approximations.
The square root of 21 is approximately 4.5826 by taking the square root of 25, which is 5, and then estimating √21 as being slightly less than 5.
The square root of 17 is approximately 4.1231 by taking the square root of 16, which is 4, and then estimating √17 as being slightly greater than 4.
Comparing these decimal approximations, we can see that 4.5826 is greater than 4.1231.
Therefore, √21 > √17.