What is the square root of 141?

Steps to Solve

1. Estimate the value

Since 141 is not a perfect square, we start by estimating its square root. We know that the square root of 141 will be between the square roots of 121 and 144, which are 11 and 12 respectively.

$$11^2 = 121 \ 12^2 = 144$$

So, $11 < \sqrt{141} < 12$.

2. Refine the estimate using the average

To refine our estimate, we take the average of 11 and 12:

$$\text{Average} = \frac{11 + 12}{2} = 11.5$$

Next, square 11.5 to see how close it gets to 141:

$$11.5^2 = 132.25$$

Since 132.25 < 141, we know that the square root of 141 is slightly greater than 11.5.

3. Further refinement by binary search (if required)

To further refine, we can test the midpoint of 11.5 and 12:

$$\text{Midpoint} = \frac{11.5 + 12}{2} = 11.75$$

Square 11.75 to see the result:

$$11.75^2 = 138.0625$$

Since 138.0625 < 141, we know the root lies between 11.75 and 12.

Next, test the midpoint of 11.75 and 12:

$$\text{Midpoint} = \frac{11.75 + 12}{2} = 11.875$$

Square 11.875:

$$11.875^2 = 141.015625$$

Since this is very close to 141, we could stop here with our estimate.