How to Calculate the Square Root of 9.8

How to Calculate the Square Root of 9.8

The square root of 9.8 can be calculated using various methods, ranging from manual computation to the use of advanced computational tools. In this article, we will explore the different techniques to find the value of the square root of 9.8, both approximately and precisely.

Calculating the Square Root of 9.8 Using a Calculator

One of the simplest and most straightforward ways to determine the square root of 9.8 is by using a calculator. If you have a scientific calculator, you can simply type sqrt(9.8) or use the square root function to get the result directly. Using such a device, the square root of 9.8 is approximately:

sqrt{9.8} ≈ 3.1305

For a more precise value, advanced computational tools or high-precision scientific calculators can provide the exact square root as:

sqrt{9.8} ≈ 3.130495168

Manual Calculation Methods

If you do not have a calculator or prefer to compute the square root manually, you can use different methods such as long division, digit-by-digit approximation, or the Newton-Raphson method. Here, we will demonstrate a simple digit-by-digit approximation technique:

Start by expressing 9.8 as a fraction:

9.8 98/10

Then, take the square root of the numerator and the denominator separately:

√9.8 √(98/10) √98 / √10

Next, approximate the square roots of 98 and 10:

√98 ≈ 9.89949; √10 ≈ 3.16228

Divide the two results to get the square root of 9.8:

√9.8 ≈ 9.89949 / 3.16228 ≈ 3.1305

Step-by-Step Explanation

To provide a more detailed understanding, let's break down the calculation into a step-by-step process:

Start by grouping the digits into pairs, starting from the decimal point:

09 . 80

For the integer part, start with the highest pair and find the largest integer whose square is less than or equal to the pair:

√9.8 √(09.80) ≈ 3 (since 32 9 and 42 16, which is too large)

Subtract the square of the integer from the pair and bring down the next pair:

9 - 32 6 . 80

Add a decimal point and a zero, and find the next digit in the quotient:

3.1 (since 312 961, which is close to 960)

Continue refining the approximation by bringing down more digits as needed:

(3.13)2 9.7969, very close to 9.8

Conclude with the final approximation:

√9.8 ≈ 3.13

This method provides a close approximation and can be used for more complex numbers as well.

Conclusion

The square root of 9.8 is approximately 3.1305, which can be refined to 3.130495168 using advanced computational tools. Whether you use a calculator or a manual method, understanding these techniques can help you accurately determine the square roots of various numbers.