# Finding the Square Root of 15

As math enthusiasts and students, finding the square root of any number is an essential part of our educational journey. In this article, we will delve into the topic of finding the square root of 15 using different methods.

To start with, the square root of a number is that value which, when multiplied by itself, gives the original number. Therefore, the square root of 15 can be represented as √15, where √ represents the symbol for a square root.

Method 1: Finding the Square Root of 15 using Prime Factorization

One way to find the square root of 15 is through prime factorization. To do this, we first need to break down 15 into its prime factors, which are 3 and 5.

15 = 3 x 5

Next, we group the prime factors in pairs, starting from the smallest possible pair. In this case, we have only one pair which consists of 3 and 5.

√15 = √(3 x 5)

We then take one number from each pair, multiply them, and place the result outside the square root symbol.

√15 = √(3 x 5) = √(15)

Thus, the result of the square root of 15 using prime factorization is √15.

Method 2: Finding the Square Root of 15 using Long Division

Another method to find the square root of 15 is using long division. It involves the following steps:

Step 1: Group the digits in pairs, starting from the rightmost digit. If there is an odd number of digits, then the leftmost digit will be grouped with a trailing zero.

For 15, we get:

15

Step 2: Starting from the leftmost pair, find a number whose square is less than or equal to the first digit in the current pair. For 15, the leftmost digit is 1, and the largest possible number whose square is less than or equal to 1 is 1.

1 | 15

Step 3: Subtract the product of the divisor (in this case, 1) and the quotient (also 1) from the current pair of digits, and bring down the next pair of digits.

1 | 15
|
1 | 05

Step 4: Double the divisor and enter it on the left. Multiply the new divisor by itself and place it underneath the current dividend. Find the largest digit that can be placed above the divisor such that the product of the divisor and the new digit is less than or equal to the current dividend.

1 | 15
1 | 05
|
2

The largest digit here is 2. Hence, we get:

1 | 15
1 | 05
2 | 10

Step 5: Repeat the process until all the digits have been brought down.

1 | 15
1 | 05
2 | 10
| 05
2 | 05

Since we have a remainder of 5, we add a decimal point after the quotient obtained so far and append a zero to the dividend. We then repeat the division process.

1 | 15.0
1 | 05
2 | 10
| 05
2 | 05
|

We continue the process until we get the desired accuracy.

1 | 15.0
1 | 05
2 | 10
| 05
2 | 05.00
| 00

Therefore, the square root of 15 using long division is 3.87298 (approximate to five decimal places).

Method 3: Finding the Square Root of 15 using a Calculator

The easiest way to find the square root of 15 is by using a calculator. Most scientific calculators have a square root button (√) that will give you the result in seconds. Here, we get:

√15 = 3.87298 (approximate to five decimal places).

Conclusion

In conclusion, finding the square root of 15 requires different methods. These methods include prime factorization, long division, and the use of a calculator. It is essential to note that while each method may have its advantages and disadvantages, they all aim to achieve the same result. Ultimately, it is up to the individual to choose the method that best suits them.