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Using a number line, what whole number and rational number to the nearest half is 97−−√97

between?(1 point)
97−−√97 is between __ and __

5 answers

To determine the range of \( \sqrt{97} \) between whole numbers and rational numbers to the nearest half, we first need to calculate \( \sqrt{97} \).

Calculating \( \sqrt{97} \):

  • We know that \( 9^2 = 81 \) and \( 10^2 = 100 \). Thus, \( \sqrt{97} \) is between 9 and 10.
  • To refine this further, we can approximate \( \sqrt{97} \) more precisely.

Calculating a closer estimate:

  • \( 9.8^2 = 96.04 \)
  • \( 9.9^2 = 97.01 \)

Thus, \( \sqrt{97} \) is between \( 9.8 \) and \( 9.9 \).

Now, when rounding to the nearest half, we have:

  • \( 9.5 < \sqrt{97} < 10 \)

Therefore, we can conclude that:

\( \sqrt{97} \) is between 9.8 and 9.9 as a rational number to the nearest half, and between 9 and 10 as whole numbers.

So the final answer is: \( \sqrt{97} \) is between 9 and 10 and 9.8 and 9.9.

Using a number line, what whole number and rational number to the nearest half is 97−−√97
between?(1 point) 97−−√97 is between and

To find the whole number and the rational number to the nearest half where \( \sqrt{97} \) is located, we can follow these steps:

  1. Estimate \( \sqrt{97} \):

    • We know \( 9^2 = 81 \) and \( 10^2 = 100 \), so \( 9 < \sqrt{97} < 10 \).
    • To refine our estimate, let's check the squares of decimals between 9 and 10:
      • \( 9.5^2 = 90.25 \) (too low)
      • \( 9.6^2 = 92.16 \) (too low)
      • \( 9.7^2 = 94.09 \) (too low)
      • \( 9.8^2 = 96.04 \) (too low)
      • \( 9.9^2 = 98.01 \) (too high)

    Hence, we can see \( 9.8 < \sqrt{97} < 9.9 \).

  2. Rounding to the nearest half:

    • The nearest half values around \( \sqrt{97} \) would be 9.5 and 10.0. Thus, \( \sqrt{97} \) rounded to the nearest half is 9.5.

  3. Conclusions:

    • The whole number between which \( \sqrt{97} \) lies is 9 and 10.
    • The rational number to the nearest half that \( \sqrt{97} \) is closest to is 9.5.

To summarize: \( \sqrt{97} \) is between 9 and 10 (whole numbers) and 9.5 and 10 (rational numbers rounded to the nearest half).

The final answer is: \( \sqrt{97} \) is between 9 and 10, specifically 9.5 and 10 for the rational number.

bot is correct if confused the answers are
9.5 and 10!

Thank you for your patience! To clarify:

  • \( \sqrt{97} \) is between 9 and 10 (as whole numbers).
  • When rounding to the nearest half, the rational number it is between is 9.5 and 10.

So, indeed, the correct completion for your original question is: \( \sqrt{97} \) is between 9.5 and 10.

I appreciate your understanding, and if you have any more questions, feel free to ask!