Square Root of 144000

The square root is the inverse of the square of the number. 144000 is not a perfect square. The square root of 144000 is expressed in both radical and exponential form. In the radical form, it is expressed as √144000, whereas (144000)^(1/2) in the exponential form. √144000 ≈ 379.4733, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.

The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers where the long-division method and approximation method are used. Let us now learn the following methods:

• Prime factorization method
   
• Long division method
   
• Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 144000 is broken down into its prime factors.

Step 1: Finding the prime factors of 144000 Breaking it down, we get 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 5 × 5 × 5: 2^6 × 3^2 × 5^3

Step 2: Now we found out the prime factors of 144000. The second step is to make pairs of those prime factors. Since 144000 is not a perfect square, therefore, the digits of the number can’t be grouped in pairs. Therefore, calculating √144000 using prime factorization is not straightforward.

The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number for the given number. Let us now learn how to find the square root using the long division method, step by step.

Step 1: To begin with, we need to group the numbers from right to left. In the case of 144000, we need to group it as 00, 40, and 144.

Step 2: Now we need to find n whose square is less than or equal to 144. We can say n as ‘12’ because 12 × 12 = 144. Now the quotient is 12 after subtracting 144-144, the remainder is 0.

Step 3: Now let us bring down 40 which is the new dividend. Add the old divisor with the same number 12 + 12, we get 24 which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 24n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 24n × n ≤ 4000. Let us consider n as 1, now 24 × 1 × 1 = 24.

Step 6: Subtract 40 from 24, the difference is 16, and the quotient is 121.

Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 1600.

Step 8: Now we need to find the new divisor that is 243 because 243 × 6 = 1458.

Step 9: Subtracting 1458 from 1600, we get the result 142.

Step 10: Now the quotient is 121.6.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal value, continue till the remainder is zero. So the square root of √144000 is approximately 379.47.

The approximation method is another method for finding the square roots, it is an easy method to find the square root of a given number. Now let us learn how to find the square root of 144000 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √144000. The smallest perfect square less than 144000 is 40000 (200^2) and the largest perfect square more than 144000 is 160000 (400^2). √144000 falls somewhere between 379 and 380.

Step 2: Now we need to apply the formula that is: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (144000 - 40000) ÷ (160000 - 40000) = 0.86. Using the formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number which is 379 + 0.47 = 379.47, so the square root of 144000 is approximately 379.47.

• Square root: A square root is the inverse of a square. Example: 4^2 = 16, and the inverse of the square is the square root, that is, √16 = 4.

• Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.

• Principal square root: A number has both positive and negative square roots. However, it is always the positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as the principal square root.

• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.

• Perfect square: A perfect square is an integer that is the square of an integer. For example, 16 is a perfect square because it is 4 squared (4 × 4).