Understanding the Square Root of 13689 Through Prime Factorization

Prime factorization is a foundational concept in mathematics, enabling us to break down any number into its prime components. This method is particularly useful for finding the square root of a number. Let's explore how prime factorization can help us determine the square root of 13689.

Prime Factorization Method

Prime factorization involves expressing a number as a product of its prime factors. For 13689, our goal is to find its prime factors and then use them to determine the square root.

Step 1: Prime Factorization of 13689

Let's begin by factorizing 13689 using prime factorization.

Starting with 2: Is 13689 divisible by 2?

No, 13689 is not divisible by 2 since it is an odd number. Next, we try 3.

Step 2: Dividing by 3

13689 ÷ 3 4563. So, we have:

13689 3 × 4563

Step 3: Further Factorization of 4563

A similar process continues:

4563 ÷ 3 1521. Thus, we have:

13689 3 × 3 × 1521, or 32 × 1521

Step 4: Factorizing 1521

Next, we factorize 1521:

1521 ÷ 3 507

507 ÷ 3 169

So, 1521 3 × 3 × 169, or 32 × 169

Step 5: Prime Factorization of 169

169 is known to be 132. Thus, the complete prime factorization of 13689 is:

13689 34 × 132

Step 2: Finding the Square Root

To find the square root, we split the powers of the prime factors by half:

(sqrt{13689} sqrt{3^4 cdot 13^2} 3^2 cdot 13 9 cdot 13 117)

Alternative Approach: Dividing by 9 First

Another approach involves dividing the number by 9 to simplify the factorization process for even numbers.

Step 1: Dividing 13689 by 9

13689 ÷ 9 1521

Step 2: Factorizing 1521 Further

1521 ÷ 3 507

507 ÷ 3 169

So, 1521 32 × 169, and since 169 132, we have:

13689 9 × 169 9 × 132

Step 3: Finding the Square Root

(sqrt{13689} sqrt{9 cdot 13^2} 3 cdot 13 39 cdot 3 117)

Conclusion

Both methods yield the same result: the square root of 13689 is 117. By utilizing prime factorization, we can break down numbers into their prime components, which allows us to simplify complex computations and find square roots efficiently.

Understanding the process of prime factorization not only aids in finding square roots but also enhances overall mathematical proficiency. It is a valuable skill for students and professionals alike, enabling them to tackle a wide range of mathematical challenges with confidence.