How to Calculate the Square Root of 21 Using the Long Division Method

Learning how to calculate the square root of a number, like 21, using the long division method can be a fascinating and practical exercise. This method is not only historically significant but also helps in understanding the foundational mathematics behind square roots. Let's walk through the process step by step.

Step 1: Setting Up the Problem

First, we need to pair the digits starting from the decimal point. For the number 21, this can be written as 21. If we wish to find more decimal places, we can add pairs of zeros after the decimal. In our problem, we will pair 21 as it is.

Estimate the largest integer:

Find the largest integer whose square is less than or equal to 21. Here, we have: 42 16 52 25 The largest integer is 4.

Step 2: Dividing

Let's proceed with the dividing process.

Calculate the first digit:

Write 4 above the line and square it:

"4^2 16"

Subtract this from 21:

21 - 16 5

After this step, you have 5 left to work with.

Step 3: Bringing Down the Next Pair

Bring down the next pair of zeros to have 500. Now, double the divisor, which is 4, to get 8.

Find the next digit:

Determine the largest digit x such that 82 ≤ 500. Testing both 6 and 5:

Testing x 6: 80 × 6 480, 480 × 6 2880, which is too large. Testing x 5: 80 × 5 400, 400 × 5 2000, which fits within 500.

Place 5 next to 4 to make it 4.5.

4.5

Step 4: Calculating

Calculate the product and subtract:

85 × 5 425

Subtract this from 500:

500 - 425 75

Now, bring down another pair of zeros to have 7500. Double the new divisor, 85, to get 170.

Step 5: Repeating

Estimate the next digit:

Find the largest digit y such that 170y2 ≤ 7500. Testing both 4 and 5:

Testing y 4: 170 × 4 680, 680 × 4 2720, which fits within 7500. Testing y 5: 170 × 5 850, 850 × 5 4250, which is too large.

Add the digit 4 to 4.5, making it 4.54.

Step 6: Calculating Again

Calculate the product and subtract:

1704 × 4 6816

Subtract this from 7500:

7500 - 6816 684

Conclusion

You can continue this process to get more decimal places. After a few iterations, the square root of 21 is approximately 4.58. This method can be extended to find square roots of other numbers with precision.

In summary, using the long division method:

sqrt{21} approx 4.58

Now you have a solid understanding of how to calculate the square root of 21 using the long division method. This technique can be incredibly useful for various mathematical and practical applications. Happy learning!