The square root of 209 lies between
13 and 14
14 and 15
15 and 16
16 and 17
The correct answer and explanation is:
The square root of 209 lies between 14 and 15.
Explanation:
To find between which two whole numbers the square root of 209 lies, we compare 209 to perfect squares of whole numbers:
- 132=16913^2 = 169
- 142=19614^2 = 196
- 152=22515^2 = 225
- 162=25616^2 = 256
- 172=28917^2 = 289
We see that:
- 142=19614^2 = 196 which is less than 209
- 152=22515^2 = 225 which is greater than 209
Therefore, the square root of 209 must be greater than 14 but less than 15.
More detailed reasoning:
The square root function is increasing and continuous. For any number xx, if n2<x<(n+1)2n^2 < x < (n+1)^2, then x\sqrt{x} lies between nn and n+1n+1.
Here, since 196<209<225196 < 209 < 225, it follows that: 14<209<1514 < \sqrt{209} < 15
Estimating the value:
To approximate 209\sqrt{209}, consider how far 209 is from 196 and 225:
- Distance from 196 to 209 is 209−196=13209 – 196 = 13
- Distance from 196 to 225 is 225−196=29225 – 196 = 29
The difference between the squares is 29, and 209 is 13 units above 196, so the square root of 209 is roughly: 14+1329≈14+0.45=14.4514 + \frac{13}{29} \approx 14 + 0.45 = 14.45
Thus, 209≈14.45\sqrt{209} \approx 14.45.
Summary:
- Since 142=196<209<225=15214^2 = 196 < 209 < 225 = 15^2, the square root of 209 lies between 14 and 15.
- A rough estimate places it near 14.45.
- This method works for any number by comparing to perfect squares and finding approximate location between consecutive integers.