Square Root of 4200

The square root is the inverse of the square of the number. 4200 is not a perfect square. The square root of 4200 is expressed in both radical and exponential form. In the radical form, it is expressed as √4200, whereas (4200)^(1/2) in the exponential form. √4200 ≈ 64.81, 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 4200 is broken down into its prime factors.

Step 1: Finding the prime factors of 4200 Breaking it down, we get 2 × 2 × 3 × 5 × 5 × 7 × 2: 2^3 × 3^1 × 5^2 × 7^1

Step 2: Now we found out the prime factors of 4200. The second step is to make pairs of those prime factors. Since 4200 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating 4200 using prime factorization is complicated without further simplification.

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 4200, we need to group it as 00 and 42.

Step 2: Now we need to find n whose square is less than or equal to 42. We can say n as ‘6’ because 6 × 6 = 36 is less than 42. Now the quotient is 6 after subtracting 36 from 42, the remainder is 6.

Step 3: Now let us bring down 00 which is the new dividend. Add the old divisor with the same number, 6 + 6 = 12, which will be our new divisor.

Step 4: The next step is finding 12n × n ≤ 600. We can consider n as 4, now 12 × 4 = 48, and 484 × 4 = 1936.

Step 5: Subtract 1936 from 6000, the difference is 4064, and the quotient is 64.

Step 6: Since the dividend is greater than the divisor, we continue the process of adding decimals and bringing down zeroes. Adding a decimal point allows us to add two zeroes to the dividend. Now the new dividend is 406400.

Step 7: We continue the process to find the new divisor and quotient until we reach an approximation.

So the square root of √4200 ≈ 64.81

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 4200 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √4200. The smallest perfect square less than 4200 is 4096 (64^2) and the largest perfect square greater than 4200 is 4225 (65^2). √4200 falls somewhere between 64 and 65.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Applying the formula: (4200 - 4096) / (4225 - 4096) = 104 / 129 = 0.8062 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 64 + 0.81 ≈ 64.81, so the square root of 4200 is approximately 64.81.

• 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 is more commonly used in practical applications.
   
• Perfect square: A perfect square is an integer that is the square of an integer. For example, 36 is a perfect square because it is 6^2.
   
• Prime factorization: The breaking down of a number into its basic prime factors. For example, the prime factorization of 4200 is 2^3 × 3^1 × 5^2 × 7^1.