# The Square Root of 144: A Guide to Finding the Answer

The square root of 144 is a mathematical problem that many people encounter during their academic journey. It is a relatively simple calculation, but it requires an understanding of the principles of multiplication, division, and square roots. In this blog post, we will explore different methods to find the square root of 144.

Method 1: Using Prime Factorization
The first method to find the square root of 144 is through prime factorization. This is a process that involves finding the prime numbers that can be multiplied together to produce the original number. In the case of 144, we can find its prime factors by dividing it by the smallest prime numbers in order: 2, 3, 5, 7, 11, and so on.

To apply this method, we start by dividing 144 by 2, which gives us 72. Then we divide 72 by 2, which gives us 36. We can continue this process until we reach a point where we cannot divide further. In this case, we end up with 2 x 2 x 2 x 2 x 3 x 3, which means that the prime factors of 144 are 2 and 3. We can then simplify this number by multiplying the two prime factors together, 2 x 2 x 2 x 2 x 3 x 3 = 144.

Once we have the prime factorization of 144, we can take the square root of each factor, square the result, and multiply all the results together to get the square root of 144. So, the square root of 144 is equal to sqrt(2) x sqrt(2) x sqrt(2) x sqrt(2) x sqrt(3) x sqrt(3) = 2 x 2 x 2 x 2 x 3 = 12.

Method 2: Using Long Division
The second method to find the square root of 144 is through long division. This method is more straightforward than the previous one but requires a bit more practice. To use this method, we start by dividing the original number 144 by another number that we know is less than or equal to its square root. We then check the quotient and the remainder and repeat the process until we reach the answer.

For example, let’s divide 144 by 12:
1. We write it as a long division problem with 12 as the divisor and 144 as the dividend:
___________
12 | 144
2. We divide 12 into the first digit of 144, which is 1. The result is 1 with a remainder of 8.
___________
12 | 144
– 12
——
24
3. We bring down the next digit of 144, which is 4, and place it next to the remainder. We then divide 12 into the two-digit number 84, which gives us 7 with a remainder of 0.
___________
12 | 144
– 12
——
24
24
4. Since the remainder is zero, we have found the correct solution. Therefore, the square root of 144 is 12.

Method 3: Using Decimal Approximation
The third method to find the square root of 144 is through decimal approximation. This method involves estimating the square root of 144 by finding the closest perfect square before and after it. We can then find the decimal approximation of the square root of 144 by dividing it by the average of the two closest perfect squares.

For instance, the closest perfect squares to 144 are 121 and 169. We can then find the average of these two numbers, which is (121 + 169) / 2 = 145 / 2 = 72.5. We can now divide 144 by 72.5 to get the decimal approximation of its square root. So, sqrt(144) = 12 (approximately).

In conclusion, there are different methods to find the square root of 144, including prime factorization, long division, and decimal approximation. While each method has its advantages and disadvantages, it is essential to understand the principles that underlie them. By using these methods, we can solve various mathematical problems that require finding the square root of a number.