Square Root of 600

The square root is the inverse of the square of the number. 600 is not a perfect square. The square root of 600 is expressed in both radical and exponential form. In the radical form, it is expressed as √600, whereas (600)^(1/2) in the exponential form. √600 ≈ 24.4949, 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 600 is broken down into its prime factors.

Step 1: Finding the prime factors of 600 Breaking it down, we get 2 x 2 x 2 x 3 x 5 x 5: 2^3 x 3^1 x 5^2

Step 2: Now we have found the prime factors of 600. The second step is to make pairs of those prime factors. Since 600 is not a perfect square, therefore the digits of the number can’t be grouped in perfect pairs. Therefore, calculating √600 using prime factorization directly is not feasible.

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 600, we need to group it as 00 and 6.

Step 2: Now we need to find n whose square is less than or equal to 6. We can say n as ‘2’ because 2^2 = 4 is less than or equal to 6. Now the quotient is 2 after subtracting 6 - 4, the remainder is 2.

Step 3: Now let us bring down 00 which is the new dividend. Add the old divisor with the same number 2 + 2 to get 4, which will be our new divisor.

Step 4: The new divisor will be 4n, and we need to find the value of n.

Step 5: The next step is finding 4n × n ≤ 200. Let us consider n as 4, now 4 x 4 x 4 = 196.

Step 6: Subtract 200 from 196, the difference is 4, and the quotient is 24.

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 400.

Step 8: Now we need to find the new divisor that is 48, because 484 x 8 = 3872.

Step 9: Subtracting 3872 from 4000 gives the result 128.

Step 10: Now the quotient is 24.8.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose there are no decimal values, continue till the remainder is zero. So the square root of √600 ≈ 24.49.

The approximation method is another method for finding 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 600 using the approximation method.

Step 1: Now we have to find the closest perfect squares around √600. The smallest perfect square less than 600 is 576 and the largest perfect square greater than 600 is 625. √600 falls somewhere between 24 and 25.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (600 - 576) ÷ (625 - 576) = 24 ÷ 49 ≈ 0.49. Using the formula we identified the decimal point of our square root. The next step is adding the initial integer value to the decimal number, which is 24 + 0.49 = 24.49. So the square root of 600 is approximately 24.49.

• 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, which 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.

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

• Prime factorization: Prime factorization is the expression of a number as the product of its prime factors. For example, the prime factorization of 600 is 2^3 x 3 x 5^2.