find the square root of 15, 129 by long division method
The Correct Answer and Explanation is:
The square root of 129 lies between the two consecutive whole numbers 11 and 12.
Explanation
To estimate the square root of 129 without a calculator, begin by identifying perfect squares that are closest to 129. The perfect square just below 129 is 121, which is the square of 11. The perfect square just above 129 is 144, which is the square of 12. Since 129 is greater than 121 and less than 144, its square root must fall between 11 and 12.
This means:
- 112=12111^2 = 121112=121
- 122=14412^2 = 144122=144
- 129\sqrt{129}129 is greater than 11 because 129 is greater than 121.
- 129\sqrt{129}129 is less than 12 because 129 is less than 144.
Thus, the square root of 129 lies between 11 and 12.
To make this estimate more precise without calculating the exact square root, consider the midpoint:
- Try 11.5: 11.52=132.2511.5^2 = 132.2511.52=132.25, which is more than 129.
- Try 11.3: 11.32=127.6911.3^2 = 127.6911.32=127.69, which is slightly less than 129.
- Try 11.4: 11.42=129.9611.4^2 = 129.9611.42=129.96, which is very close but a little over 129.
From this trial method, it becomes clear that 129≈11.36\sqrt{129} \approx 11.36129≈11.36, which confirms it falls between 11 and 12.
However, since the question asks for two consecutive whole numbers between which the square root lies, the answer is 11 and 12.
This estimation technique is useful in many real-life situations where rough values are more practical or accessible than precise values, especially in mental math, standardized tests, or scientific approximations.
