Square Root of 5648

The square root is the inverse of the square of the number. 5648 is not a perfect square. The square root of 5648 is expressed in both radical and exponential form. In the radical form, it is expressed as √5648, whereas (5648)¹/² in the exponential form. √5648 ≈ 75.146, 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 5648 is broken down into its prime factors.

Step 1: Finding the prime factors of 5648 Breaking it down, we get 2 x 2 x 2 x 2 x 353: 2⁴ x 353

Step 2: Now we found out the prime factors of 5648. The second step is to make pairs of those prime factors. Since 5648 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 5648 using prime factorization is impossible.

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 5648, we need to group it as 48 and 56.
 

Step 2: Now we need to find n whose square is closest to or less than 56. We can use n as ‘7’ because 7 x 7 is 49, which is less than 56. Now the quotient is 7, and after subtracting 49 from 56, the remainder is 7.

Step 3: Now, let us bring down 48, which is the new dividend. Add the old divisor with the same number (7 + 7) to get 14, which will be our new divisor.

Step 4: The new divisor will be the sum of the current divisor and the next digit of the quotient. Now we get 14n as the new divisor, and we need to find the value of n.

Step 5: The next step is to find 14n x n ≤ 748. Let us consider n as 5 since 145 x 5 = 725, which is closest to 748.

Step 6: Subtract 725 from 748; the difference is 23, and the quotient is 75.

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 2300.

Step 8: Now we need to find the new divisor. We find 150n x n ≤ 2300. Let us consider n as 1 since 1501 x 1 = 1501.

Step 9: Subtracting 1501 from 2300, we get 799.

Step 10: 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 √5648 ≈ 75.146.

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 5648 using the approximation method.

Step 1: Now we have to find the closest perfect square of √5648.

The smallest perfect square less than 5648 is 5476 (74²), and the largest perfect square greater than 5648 is 5776 (76²). √5648 falls somewhere between 74 and 76.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Using the formula (5648 - 5476) ÷ (5776 - 5476) ≈ 0.146.

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 74 + 0.146 ≈ 74.146.

So the square root of 5648 is approximately 75.146.

• Square root: A square root is the inverse of a square. Example: 4² = 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.

• Approximation method: A mathematical approach used to find a close estimate of the square root for non-perfect squares.

• Long division method: A method used to find the square root of a number by dividing it into pairs of digits and using iterative division steps.