Square Root of 1200

The square root is the inverse of the square of the number. 1200 is not a perfect square. The square root of 1200 is expressed in both radical and exponential form. In the radical form, it is expressed as √1200, whereas (1200)^(1/2) in the exponential form. √1200 ≈ 34.641, 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 1200 is broken down into its prime factors.

Step 1: Finding the prime factors of 1200 Breaking it down, we get 2 × 2 × 2 × 2 × 3 × 5 × 5: 2^4 × 3 × 5^2

Step 2: Now we found out the prime factors of 1200. The second step is to make pairs of those prime factors. Since 1200 is not a perfect square, therefore, the digits of the number can’t be grouped in pairs completely.

Calculating √1200 using prime factorization results in 2 × 5 × √3 = 10√3.

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 1200, we need to group it as 00 and 12.

Step 2: Now we need to find n whose square is less than or equal to 12. We can say n is ‘3’ because 3 × 3 = 9, which is less than or equal to 12. Now the quotient is 3, after subtracting 9 from 12, the remainder is 3.

Step 3: Now let us bring down 00, making the new dividend 300. Add the old divisor with the same number, 3 + 3, we get 6, which will be our new divisor prefix.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 6n as the new divisor prefix, we need to find the value of n.

Step 5: The next step is finding 6n × n ≤ 300. Let us consider n as 5, now 65 × 5 = 325, which is more than 300, so we try n as 4.

Step 6: With n as 4, 64 × 4 = 256, which is less than or equal to 300. Subtract 256 from 300; the difference is 44, and the quotient is 34.

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 4400.

Step 8: Now we need to find the new divisor. We put 68 as the prefix, making it 684, and find n so that 684n × n ≤ 4400.

Step 9: Using n as 6, 684 × 6 = 4104, which is less than 4400. Subtracting 4104 from 4400 gives a remainder.

Step 10: Now the quotient is 34.6.

Step 11: Continue these steps until we get the desired accuracy.

The approximate square root of √1200 is 34.641.

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 1200 using the approximation method.

Step 1: Now we have to find the closest perfect squares around 1200. The smallest perfect square of 1200 is 1156 (34^2) and the largest perfect square of 1200 is 1225 (35^2). √1200 falls somewhere between 34 and 35.

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 (1200 - 1156) / (1225 - 1156) = 44 / 69 ≈ 0.64. 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 34 + 0.64 = 34.64, so the square root of 1200 is approximately 34.64.

• 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 usually the positive square root that is used due to its applications in the real world. That is why it is also known as the principal square root.
   
• Prime factorization: The process of expressing a number as the product of its prime factors. For example, the prime factorization of 1200 is 2^4 × 3 × 5^2.
   
• 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.