Answer :
Sure, let's solve the problem of finding which number is closest to [tex]\(-\sqrt{104}\)[/tex].
### Step-by-Step Solution:
1. Calculate the Square Root:
First, we need to estimate the value of [tex]\(\sqrt{104}\)[/tex]. The value of [tex]\(\sqrt{104}\)[/tex] is approximately 10.198.
2. Negate the Value:
Since we are interested in [tex]\(-\sqrt{104}\)[/tex], we take the negative of this value:
[tex]\[
-\sqrt{104} \approx -10.198
\][/tex]
3. Compare with Given Choices:
The next step is to compare [tex]\(-10.198\)[/tex] to the given choices:
- 11.1
- -10.1
- -11.1
- 10.1
4. Find the Closest Number:
We need to determine which number is closest to [tex]\(-10.198\)[/tex]. We can do this by checking the absolute differences between [tex]\(-10.198\)[/tex] and each of the choices:
- Difference with 11.1:
[tex]\[
|11.1 - (-10.198)| = 11.1 + 10.198 = 21.298
\][/tex]
- Difference with -10.1:
[tex]\[
|-10.1 - (-10.198)| = |-10.1 + 10.198| = 0.098
\][/tex]
- Difference with -11.1:
[tex]\[
|-11.1 - (-10.198)| = |-11.1 + 10.198| = 0.902
\][/tex]
- Difference with 10.1:
[tex]\[
|10.1 - (-10.198)| = 10.1 + 10.198 = 20.298
\][/tex]
5. Conclusion:
Among the provided options, the number [tex]\(-10.1\)[/tex] has the smallest difference (0.098) when compared to [tex]\(-10.198\)[/tex]. Therefore, [tex]\(-10.1\)[/tex] is the closest number to [tex]\(-\sqrt{104}\)[/tex].
[tex]\[
\boxed{-10.1}
\][/tex]
This is the number closest to [tex]\(-\sqrt{104}\)[/tex].
### Step-by-Step Solution:
1. Calculate the Square Root:
First, we need to estimate the value of [tex]\(\sqrt{104}\)[/tex]. The value of [tex]\(\sqrt{104}\)[/tex] is approximately 10.198.
2. Negate the Value:
Since we are interested in [tex]\(-\sqrt{104}\)[/tex], we take the negative of this value:
[tex]\[
-\sqrt{104} \approx -10.198
\][/tex]
3. Compare with Given Choices:
The next step is to compare [tex]\(-10.198\)[/tex] to the given choices:
- 11.1
- -10.1
- -11.1
- 10.1
4. Find the Closest Number:
We need to determine which number is closest to [tex]\(-10.198\)[/tex]. We can do this by checking the absolute differences between [tex]\(-10.198\)[/tex] and each of the choices:
- Difference with 11.1:
[tex]\[
|11.1 - (-10.198)| = 11.1 + 10.198 = 21.298
\][/tex]
- Difference with -10.1:
[tex]\[
|-10.1 - (-10.198)| = |-10.1 + 10.198| = 0.098
\][/tex]
- Difference with -11.1:
[tex]\[
|-11.1 - (-10.198)| = |-11.1 + 10.198| = 0.902
\][/tex]
- Difference with 10.1:
[tex]\[
|10.1 - (-10.198)| = 10.1 + 10.198 = 20.298
\][/tex]
5. Conclusion:
Among the provided options, the number [tex]\(-10.1\)[/tex] has the smallest difference (0.098) when compared to [tex]\(-10.198\)[/tex]. Therefore, [tex]\(-10.1\)[/tex] is the closest number to [tex]\(-\sqrt{104}\)[/tex].
[tex]\[
\boxed{-10.1}
\][/tex]
This is the number closest to [tex]\(-\sqrt{104}\)[/tex].
