Question

Using a number line, what whole number and rational number to the nearest half is 58−−√ between?

Answer 1

Using rational approximations, what is an inequality statement comparing 81−−√
and 8–√8
? Enter <, >, or =.

Answer 2

Using rational approximations, what is an inequality statement comparing 81−−√
and 88–-√
?

Answer 3

To determine the whole number and rational number to the nearest half that \( \sqrt{58} \) is between, we first need to calculate \( \sqrt{58} \).

We know that:

- \( 7^2 = 49 \)
- \( 8^2 = 64 \)

Since \( 49 < 58 < 64 \), it follows that:

\[
7 < \sqrt{58} < 8
\]

Next, to find a more precise value for \( \sqrt{58} \), we can use a calculator:

\[
\sqrt{58} \approx 7.6158
\]

Now, we will determine which whole number and rational number to the nearest half \( \sqrt{58} \) falls between.

1. The whole number:
- Since \( 7 < \sqrt{58} < 8 \), the whole number that \( \sqrt{58} \) is between is 7 and 8.

2. The rational number to the nearest half:
- The nearest halves around \( 7.6158 \) are \( 7.5 \) and \( 8.0 \). Therefore, \( \sqrt{58} \) is closer to \( 7.5 \), but since it is more than \( 7.5 \), we can say it is between \( 7.5 \) and \( 8.0 \).

Thus, we conclude that \( \sqrt{58} \) is between the whole number 7 and the rational number \( 7.5 \) (to the nearest half).

In summary:
- Whole number: 7
- Rational number (to the nearest half): 7.5

Answer 4

To compare \( \sqrt{81} \) and \( 8 - \sqrt{8} \), let’s calculate both expressions:

1. **Calculate \( \sqrt{81} \):**
\[
\sqrt{81} = 9
\]

2. **Calculate \( 8 - \sqrt{8} \):**
First, we need to find \( \sqrt{8} \). We can estimate \( \sqrt{8} \):
\[
\sqrt{8} \approx 2.828 \text{ (since } 2.828^2 \approx 8\text{)}
\]

Now, substitute this value into \( 8 - \sqrt{8} \):
\[
8 - \sqrt{8} \approx 8 - 2.828 = 5.172
\]

3. **Compare the two results:**
We have:
\[
\sqrt{81} = 9
\]
\[
8 - \sqrt{8} \approx 5.172
\]

Now, we can write the inequality:
\[
9 > 5.172
\]

Thus, we conclude:
\[
\sqrt{81} > 8 - \sqrt{8}
\]

The final inequality statement is:
\[
\sqrt{81} > 8 - \sqrt{8}
\]

Answer 5

To compare \( \sqrt{81} \) and \( \sqrt{88} \), let's calculate both values.

1. **Calculate \( \sqrt{81} \):**
\[
\sqrt{81} = 9
\]

2. **Calculate \( \sqrt{88} \):**
We can estimate \( \sqrt{88} \). First, notice that:
\[
9^2 = 81 \quad \text{and} \quad 10^2 = 100
\]
This means:
\[
9 < \sqrt{88} < 10
\]

To refine this approximation, we can calculate \( \sqrt{88} \) approximately as follows:
\[
\sqrt{88} \approx 9.38 \quad (\text{since } 9.38^2 \approx 88)
\]

3. **Compare the two results:**
Now we have:
\[
\sqrt{81} = 9
\]
\[
\sqrt{88} \approx 9.38
\]

4. **Determine the inequality:**
Since \( 9 < 9.38 \), we can write:
\[
\sqrt{81} < \sqrt{88}
\]

Thus, the final inequality statement is:
\[
\sqrt{81} < \sqrt{88}
\]