Square Root of 4224

The square root is the inverse of the square of a number. 4224 is not a perfect square. The square root of 4224 is expressed in both radical and exponential form. In radical form, it is expressed as √4224, whereas in exponential form it is expressed as (4224)^(1/2). √4224 ≈ 65.008, 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, for non-perfect square numbers, 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 4224 is broken down into its prime factors.

Step 1: Finding the prime factors of 4224 Breaking it down, we get 2^6 × 3 × 11

Step 2: Now that we have found the prime factors of 4224, the next step is to make pairs of those prime factors. Since 4224 is not a perfect square, the digits of the number can’t be grouped into pairs.

Therefore, calculating 4224 using prime factorization results in an approximate value.

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 4224, we need to group it as 42 and 24.

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

Step 3: Bring down 24 to make the new dividend 624. Add the old divisor (6) with itself to get 12, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 12n as the new divisor, we need to find the value of n.

Step 5: Find 12n × n ≤ 624. Let n be 5; 125 × 5 = 625, which is too large.

Step 6: Try n as 4, 124 × 4 = 496.

Step 7: Subtract 496 from 624 to get 128, and the quotient is 64.

Step 8: Since the dividend is less than the divisor, add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. The new dividend is 12800.

Step 9: The new divisor is 1289 because 1289 × 9 = 11601.

Step 10: Subtracting 11601 from 12800 gives 1199 as the remainder.

Step 11: Continue these steps until we get two numbers after the decimal point.

The square root of √4224 is approximately 65.008.

The approximation method is another way to find 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 4224 using the approximation method.

Step 1: We have to find the closest perfect squares around √4224. The smallest perfect square less than 4224 is 4096 (which is 64^2), and the largest perfect square more than 4224 is 4225 (which is 65^2). √4224 falls somewhere between 64 and 65.

Step 2: Now apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Using the formula: (4224 - 4096) ÷ (4225 - 4096) = 128 ÷ 129 = 0.992. Adding the decimal to the integer part: 64 + 0.992 ≈ 64.992, so the square root of 4224 is approximately 65.008.

• 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 means √16 = 4.
   
• Radical form: Expressing the square root in the form of a symbol, like √4224.
   
• Long division method: A method used to find the square roots of numbers that are not perfect squares.
   
• Irrational number: A number that cannot be written as a simple fraction; its decimal form does not terminate or repeat.
   
• Prime factorization: Breaking down a composite number into the product of its prime factors, used to simplify expressions.