Square Root Of 289 By Division Method

Finding the Square Root of 289 Using the Division Method: A Comprehensive Guide

The square root of a number is a value that, when multiplied by itself, gives the original number. Finding the square root can be approached in several ways, but the division method offers a systematic and relatively straightforward approach, especially for larger numbers. This comprehensive guide will walk you through the process of calculating the square root of 289 using the division method, explaining each step in detail. We'll also delve into the underlying principles and explore why this method works.

Understanding the Division Method for Square Roots

The division method, also known as the long division method, for finding square roots is an iterative process. It involves grouping digits, estimating the square root digit by digit, and refining the estimate through successive subtractions and divisions. While seemingly complex at first glance, the method is built upon simple arithmetic operations and follows a consistent pattern.

Before we dive into the calculation, let's establish a foundational understanding of the method's logic. The core principle lies in leveraging the binomial expansion of (a+b)² = a² + 2ab + b². We use this expansion to systematically subtract perfect squares from the original number until we reach zero or a negligible remainder.

Step-by-Step Calculation of √289

Let's apply the division method to calculate the square root of 289.

Step 1: Grouping the Digits

The first step involves grouping the digits of the number (289) in pairs, starting from the decimal point (or the rightmost digits in the case of whole numbers). For 289, we have one group: 2 89.

Step 2: Finding the First Digit of the Square Root

Now, we find the largest integer whose square is less than or equal to the leftmost group (2 in this case). The integer is 1 (since 1² = 1 < 2, while 2² = 4 > 2). This becomes the first digit of our square root. We subtract the square of this digit (1²) from the first group (2).

2 89 | 1
-1
---
1

Step 3: Bringing Down the Next Pair

Next, we bring down the next pair of digits (89). This forms the new dividend (189).

2 89 | 1
-1
---
1 89

Step 4: Doubling the Current Quotient and Adding a Placeholder

Double the current quotient (which is 1), giving us 2. We then append a placeholder digit 'x' next to this doubled quotient (2x), where 'x' represents a digit we need to determine. This forms the divisor (2x).

Step 5: Determining the Next Digit

Now, we need to find the digit 'x' such that when we multiply (2x) by x, the result is less than or equal to the dividend (189). Let's try different values of x:

• If x = 7: (27) * 7 = 189. This works perfectly!

Therefore, 'x' is 7, and it becomes the next digit of our square root. We subtract the product (189) from the dividend (189).

2 89 | 17
-1
---
1 89
-1 89
---
0

Step 6: Final Result

The remainder is 0, indicating that the square root of 289 is exactly 17.

Verification:

To verify our result, we can simply multiply 17 by itself: 17 * 17 = 289. This confirms our calculation.

Why Does This Method Work? A Deeper Dive into the Math

The division method's effectiveness stems from the binomial expansion of (a + b)², as mentioned earlier. In our example:

• a represents the initial estimate (10 in this case, as we start with the tens digit).
• b represents the next digit to be determined (7 in this case).

(a + b)² = a² + 2ab + b²

The method systematically subtracts the terms of this expansion:

1. a²: Initially, we subtract 1² (which is a² for the first iteration) from 2.
2. 2ab + b²: This is represented by the subtraction of (2x) * x in step 5, where '2x' is essentially 2a + b.

By successively subtracting these terms, we progressively refine our estimate of the square root until the remainder is zero (or negligible for non-perfect squares).

Handling Numbers with More Digits

The division method can be extended to handle numbers with more digits. The process remains the same, but it involves more iterations. The key steps remain consistent: grouping digits in pairs, finding the first digit, bringing down pairs, doubling the quotient, adding a placeholder, and determining the next digit.

For example, let's consider finding the square root of 12321.

1. Grouping: 1 23 21
2. First Digit: The square root of 1 is 1. (1² = 1)
3. Subtraction and Bring Down: 123 -1 = 123.
4. Double Quotient + Placeholder: 2x * x where x needs to be determined.
5. Determine Next Digit: We need a value for x such that (2x) * x ≤ 123. If we test several values:
   • If x = 1: 21 * 1 = 21
   • If x = 2: 22 * 2 = 44
   • If x = 3: 23 * 3 = 69
   • If x = 4: 24 * 4 = 96
   • If x = 5: 25 * 5 = 125 (Too large, so we chose 4)
6. Subtraction: 123 - 96 = 27
7. Bring down next pair: 2721
8. Double the Quotient + Placeholder: 14 * 10x, let's try x = 1
9. Determine the Next Digit: 841, subtract from 2721 leaves 0, so the last digit is 1
10. Final result: 111

This method ensures that even for larger numbers, we can systematically and accurately approximate (or find the exact value of) their square roots.

Conclusion: Mastering the Division Method

The division method for finding square roots provides a powerful and systematic technique that can be applied to a wide range of numbers. While it may appear complex initially, understanding the underlying principles of binomial expansion and the step-by-step procedure makes the method accessible and readily applicable. This detailed guide has demonstrated the technique and its underlying mathematical principles, equipping you to confidently calculate square roots using this effective method. Practice is key to mastering the method, so try applying it to different numbers to build your proficiency and understanding. Remember, with consistent practice, calculating square roots using the division method will become second nature.