Find two consecutive whole numbers that the square root of 129 lies in between
The Correct Answer and Explanation is:
Correct Answer:
The square root of 129 lies between 11 and 12.
Explanation:
To find two consecutive whole numbers that the square root of 129 lies between, consider perfect squares that are close to 129.
Start with:
- 112=12111^2 = 121112=121
- 122=14412^2 = 144122=144
Since 129 is greater than 121 but less than 144, its square root must be more than 11 but less than 12. This means the square root of 129 lies between 11 and 12.
To check how close 129 is to each square:
- 129−121=8129 – 121 = 8129−121=8
- 144−129=15144 – 129 = 15144−129=15
This shows that 129 is closer to 121 than to 144. Therefore, the square root of 129 is slightly more than 11 but not close enough to reach 12.
Now estimate the square root more precisely by testing values between 11 and 12:
- 11.32=127.6911.3^2 = 127.6911.32=127.69
- 11.42=129.9611.4^2 = 129.9611.42=129.96
These calculations show that the square root of 129 is between 11.3 and 11.4.
- 11.362=129.049611.36^2 = 129.049611.362=129.0496
So, 129≈11.36\sqrt{129} \approx 11.36129≈11.36, which confirms that the value lies between 11 and 12.
Whole numbers are numbers without fractions or decimals. Since the square root of 129 is not a whole number, the closest whole numbers that surround it are 11 and 12.
This process demonstrates how square roots of non-perfect squares can be approximated using nearby perfect squares and successive estimation. Understanding square roots helps in many applications including geometry, engineering, and physics, where length and distance are calculated. Knowing how to place irrational numbers between whole numbers is a basic but important skill in number sense and algebra.
