How to Find the Square Root of 13689 and Other Numbers

Introduction to Square Roots

In mathematics, the square root of a number is a value that, when multiplied by itself, gives the original number. The square root has numerous applications in various fields such as physics, engineering, and data analysis. This article will walk you through the process of finding the square root of 13689, and we'll explore the general method to solve similar problems.

Finding the Square Root of 13689

Let's start by finding the square root of 13689. We can do this by breaking down the number into groups of two digits from the right, and then applying a few simple rules.

Step 1: Grouping Digits

First, we group the digits of 13689 as 136 and 89.

Step 2: Finding the Squared Digits

Next, we identify the largest perfect square less than or equal to 136. Here, that would be 121 (which is (11^2)). We also note that the next square is 144 (which is (12^2)).

Step 3: Refining the Answer

Since 144 is more than 136, the unit digit of the square root of 13689 must be a digit whose square ends in 89. The only digits that fit this criterion are 3 and 7.

Step 4: Exact Calculation

Now, we need to check both possibilities: 113 and 117. Squaring 113, we get:

[113^2 12769]

Squaring 117, we get:

[117^2 13689]

Thus, the square root of 13689 is 117.

Verification

We can verify our answer by squaring 117:

[117 times 117 13689]

This confirms that 117 is indeed the square root of 13689.

Applying the Method to Other Numbers

Let's apply this method to find the square root of 134789 and 12100.

Number: 134789

Group the digits as 134 and 789. The perfect square less than or equal to 134 is 121, and 144 is the next square.

The unit digit of the square root must be a digit whose square ends in 89. The only digits that fit are 9 and 7. Since 7 is closer to 9, we might start with 369 or 377. By further calculation, we get:

3737 13689 (incorrect)

Therefore, the square root of 134789 is approximately 367.035.

Number: 12100

Group the digits as 121 and 00. The perfect square less than or equal to 121 is 121, which is (11^2).

We know that:

[110^2 12100]

So, the square root of 12100 is 110.

Conclusion

Understanding how to find the square root of a number is not just an academic exercise; it is a valuable skill with practical applications. By breaking down the problem and applying logical steps, we can efficiently find square roots without the need for a calculator.

Now, let's summarize the key steps to find the square root of any number:

Group the digits of the number into pairs from the right. Identify the largest perfect square less than or equal to the first group. Determine the unit digit of the square root based on the last digit of the square. Verify the answer by squaring the calculated value.

With practice, finding square roots becomes a straightforward process.

For further exploration, consider delving into more complex number theory topics or exploring algorithms for more efficient square root calculations. Happy calculating!