To find the approximate value of [tex]\(\sqrt{103}\)[/tex] and to locate it on a number line, we can follow these steps:
1. Understanding the Problem:
- We need to determine the square root of 103.
- The value should be given in decimal form.
- The result should be rounded to the nearest hundredth.
2. Estimating the Square Root:
- We first determine which two perfect squares 103 lies between. The perfect squares close to 103 are 100 (with a square root of 10) and 121 (with a square root of 11).
- Hence, [tex]\(\sqrt{103}\)[/tex] is between 10 and 11.
3. Calculating the Exact Value:
- On refining our calculation, the square root of 103 is found to be 10.14889156509222.
4. Rounding to the Nearest Hundredth:
- We look at the third decimal place to decide whether to round up or down. Since the digit in the third decimal place is 8, which is greater than 5, we round up.
- Thus, rounding 10.14889156509222 to the nearest hundredth gives us 10.15.
5. Conclusion:
- Therefore, [tex]\(\sqrt{103}\)[/tex] is approximately 10.15 when rounded to the nearest hundredth.
So, the value of [tex]\(\sqrt{103}\)[/tex], when placed on a number line, and rounded to the nearest hundredth, is approximately [tex]\( \boxed{10.15} \)[/tex].
