Question
Approximate Values of Irrational Numbers Practice
Complete this assessment to review what you've learned. It will not count toward your grade.
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Question
Using a number line, what whole number and rational number to the nearest half is 97−−√
between?(1 point)
Answers
GPT-4o mini
To determine the whole number and the rational number (to the nearest half) that \(\sqrt{97}\) falls between, we can start by estimating the value of \(\sqrt{97}\).
1. **Identify the perfect squares around 97**:
- \(9^2 = 81\)
- \(10^2 = 100\)
From this, we know that:
\[
9 < \sqrt{97} < 10
\]
2. **Finding a more precise estimate**:
- We can start calculating some squares to find a value closer to 97.
- \(9.5^2 = 90.25\)
- \(9.7^2 = 94.09\)
- \(9.8^2 = 96.04\)
- \(9.9^2 = 98.01\)
Since \(9.8^2\) is just below 97 and \(9.9^2\) is just above 97, we can say:
\[
9.8 < \sqrt{97} < 9.9
\]
3. **Rational number to the nearest half**:
- The nearest halves around \(9.8\) would be \(9.5\) and \(10.0\).
- Since \(\sqrt{97} \approx 9.848\), it rounds to \(9.5\).
In conclusion, \(\sqrt{97}\) is between the whole number \(9\) and the rational number \(9.5\) (to the nearest half).
So, your answer is that \(\sqrt{97}\) is between **9 and 9.5**.
1. **Identify the perfect squares around 97**:
- \(9^2 = 81\)
- \(10^2 = 100\)
From this, we know that:
\[
9 < \sqrt{97} < 10
\]
2. **Finding a more precise estimate**:
- We can start calculating some squares to find a value closer to 97.
- \(9.5^2 = 90.25\)
- \(9.7^2 = 94.09\)
- \(9.8^2 = 96.04\)
- \(9.9^2 = 98.01\)
Since \(9.8^2\) is just below 97 and \(9.9^2\) is just above 97, we can say:
\[
9.8 < \sqrt{97} < 9.9
\]
3. **Rational number to the nearest half**:
- The nearest halves around \(9.8\) would be \(9.5\) and \(10.0\).
- Since \(\sqrt{97} \approx 9.848\), it rounds to \(9.5\).
In conclusion, \(\sqrt{97}\) is between the whole number \(9\) and the rational number \(9.5\) (to the nearest half).
So, your answer is that \(\sqrt{97}\) is between **9 and 9.5**.