Square Root of 5248

The square root is the inverse of the square of the number. 5248 is not a perfect square. The square root of 5248 is expressed in both radical and exponential form. In the radical form, it is expressed as √5248, whereas (5248)^(1/2) in the exponential form. √5248 ≈ 72.44493, 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 like 5248, 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 5248 is broken down into its prime factors.

Step 1: Finding the prime factors of 5248

Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 2 x 41: 2^6 x 41

Step 2: Now we found out the prime factors of 5248. The second step is to make pairs of those prime factors. Since 5248 is not a perfect square, the digits of the number can’t be grouped into pairs completely. Therefore, calculating 5248 using prime factorization directly is not possible.

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 5248, we need to group it as 48 and 52.

Step 2: Now we need to find n whose square is closest to 52. We can say n as ‘7’ because 7 x 7 = 49, which is less than 52. Now the quotient is 7, and after subtracting 49 from 52, the remainder is 3.

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

Step 4: The new divisor will be 14n. We need to find the value of n such that 14n x n ≤ 348.

Step 5: By trial, finding 14 x 2 x 2 = 56, we see 14 x 2 = 28 is too small. Trying 14 x 5 = 70, we find 70 x 5 = 350, slightly over. So, 70 x 4 = 280 fits.

Step 6: Subtract 280 from 348, the difference is 68, and the quotient becomes 74.

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 6800.

Step 8: Now we need to find the new divisor that is 149 because 149 x 4 = 596, and 149 x 5 = 745, so 149 x 4 fits closest.

Step 9: Subtracting 596 from 6800, we get the result 204.

Step 10: Continue doing these steps until we get two numbers after the decimal point.

So the square root of √5248 is approximately 72.44.

The approximation method is another method for finding square roots, and 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 5248 using the approximation method.

Step 1: Now we have to find the closest perfect square of √5248. The smallest perfect square less than 5248 is 4900 and the largest perfect square greater than 5248 is 5625. √5248 falls somewhere between 70 and 75.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (5248 - 4900) ÷ (5625 - 4900) = 348 ÷ 725 ≈ 0.48. 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 70 + 0.48 = 70.48, so the square root of 5248 is approximately 72.44.

• Square root: A square root is the inverse of a square. For 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.

• Approximation: Approximating involves estimating a number to a suitable degree of accuracy. For example, √2 ≈ 1.414.

• 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 4^2.

• Exponent: An exponent refers to the number of times a number is multiplied by itself. For example, in 2^3, 3 is the exponent, indicating that 2 is multiplied by itself three times.