Which point best approximates the square root of 123?
The correct answer and explanation is:
To approximate the square root of 123, we begin by considering perfect squares close to 123. The square root of 121 is 11, and the square root of 144 is 12. Therefore, we know that √123 lies between 11 and 12.
To get a more precise approximation, we can use a simple estimation method like averaging or trial and error. First, let’s estimate by averaging 11 and 12. The average is:
(11 + 12) / 2 = 11.5
Now, let’s square 11.5 to see if it’s closer to 123:
11.5² = 132.25
Since 132.25 is greater than 123, it means our estimate is too high. Next, try a slightly smaller number, say 11.1:
11.1² = 123.21
This value is closer to 123, so √123 is approximately 11.1. To refine this further, let’s check 11.05:
11.05² = 122.1025
This value is just below 123, suggesting that the square root of 123 is closer to 11.1 but slightly less than that. Therefore, 11.1 is a good approximation for the square root of 123, and it falls between 11.05 and 11.1.
This method of approximation is useful for estimating square roots when exact values are not easily obtainable. In practical scenarios, further decimal places might be needed, but for most purposes, 11.1 is a sufficient approximation.