Introduction

While the square root of 1.2 can be calculated using a calculator, understanding the process behind it can greatly enhance your mathematical skills. This article explores various methods to estimate and find the square root of 1.2 accurately.

Using a Calculator

For a quick and precise answer, use a calculator. Input the expression √1.2 and the result will be approximately 1.0954. This method is straightforward but lacks the educational value of manual estimation methods.

Manual Estimations

Identifying Perfect Squares

The perfect squares closest to 1.2 are 1^2 1 and 2^2 4. Since 1.2 lies between 1 and 4, the square root must be between 1 and 2.

You can further refine this estimate by testing values close to 1.1 and 1.05:

1.1^2 1.21 (too high) 1.05^2 1.1025 (too low)

Based on these estimations, the square root of 1.2 is likely around 1.095.

Newton's Method

Newton's method offers a more precise calculation. Start with an initial guess x_0 1. Apply the formula:

x_{n 1} frac{1}{2}left(x_n frac{1.2}{x_n}right)

Iterate this process until the value stabilizes. Following this method will lead you to approximately 1.0954.

Simplified Estimation Method

This method uses a series of steps:

Multiply both sides of the equation by 10^2. Recognize that the square root of 120 is just below 11, so the square root of 1.2 is just below 1.1. Continue multiplying and squaring, estimating: 10^2 12000, find a number close to 118.81, which is close to 109^2. Further refine by checking 1095^2, 1096^2, and so on, until you get a closer estimate.

This method, while not as speedy, is effective and educational for deepening understanding.

Conclusion

The square root of 1.2 is approximately 1.0954. Whether you use a calculator or a manual method, the key is understanding the process and knowing when to stop based on the required level of precision.