Square Root of 4608

The square root is the inverse of the square of the number. 4608 is not a perfect square. The square root of 4608 is expressed in both radical and exponential form. In the radical form, it is expressed as √4608, whereas \(4608^{1/2}\) in the exponential form. √4608 ≈ 67.856, 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 4608 is broken down into its prime factors.

Step 1: Finding the prime factors of 4608

Breaking it down, we get 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 2.

Step 2: Now we have found the prime factors of 4608. The next step is to make pairs of those prime factors. Since 4608 is not a perfect square, the digits of the number can’t be grouped in pairs to result in a whole number as the square root. Therefore, calculating √4608 using prime factorization gives 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 4608, we need to group it as 08 and 46.

Step 2: Now we need to find n whose square is less than or equal to 46. We can say n as ‘6’ because 6 × 6 = 36, which is less than 46. Now the quotient is 6 and after subtracting 36 from 46, the remainder is 10.

Step 3: Now let us bring down 08, making the new dividend 1008. Add the old divisor with the same number, 6 + 6 = 12, which will be our new divisor.

Step 4: We need to find n such that 12n × n ≤ 1008. Let n be 8, so 128 × 8 = 1024, which is greater than 1008, so we use n = 7 instead.

Step 5: Subtract 128 × 7 = 896 from 1008, the difference is 112, and the new quotient is 67.

Step 6: 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 11200.

Step 7: Now we need to find the new divisor and continue the division process until sufficient decimal places are achieved.

So the square root of √4608 ≈ 67.856.

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 4608 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √4608. The smallest perfect square less than 4608 is 4096, and the largest perfect square greater than 4608 is 4624. √4608 falls somewhere between 64 and 68.

Step 2: Now we need to apply the approximation formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) (4608 - 4096) / (4624 - 4096) = 512 / 528 ≈ 0.9697 Using the formula, we identified the decimal point of our square root. The next step is adding the integer value we got initially to the decimal number which is 64 + 3.856 = 67.856, so the square root of 4608 is approximately 67.856.

• 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.

• Prime factorization: This is the process of expressing a number as the product of its prime factors. For example, the prime factorization of 18 is 2 × 3 × 3.

• Approximation method: A method used to find an approximate value of a square root by using values of known perfect squares close to the given number.