Square Root of 5300

The square root is the inverse of the square of the number. 5300 is not a perfect square. The square root of 5300 is expressed in both radical and exponential form. In radical form, it is expressed as √5300, whereas (5300)^(1/2) in exponential form. √5300 ≈ 72.8011, 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 5300 is broken down into its prime factors.

Step 1: Finding the prime factors of 5300

Breaking it down, we get 2 x 2 x 5 x 5 x 53: 2^2 x 5^2 x 53

Step 2: Now we have found out the prime factors of 5300. The second step is to make pairs of those prime factors. Since 5300 is not a perfect square, the digits of the number can’t be completely grouped in pairs. Therefore, calculating √5300 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 5300, we need to group it as 00 and 53.

Step 2: Now we need to find n whose square is less than or equal to 53. We can say n is ‘7’ because 7 x 7 = 49, which is less than 53. The quotient is 7, and after subtracting 49 from 53, the remainder is 4.

Step 3: Now let us bring down 00, making the new dividend 400. Add 7 (the last digit of the quotient) to the divisor, making it 14.

Step 4: The new divisor will be 14n. We need to find the value of n such that 14n x n ≤ 400. Let us consider n as 2, so 142 x 2 = 284.

Step 5: Subtracting 284 from 400, the difference is 116.

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 11600.

Step 7: Now we need to find the new divisor, which is 145 (since 142 + 3 = 145) because 145 x 8 = 1160.

Step 8: Subtracting 1160 from 11600 gives us 0.

Step 9: The quotient is 72.8.

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 √5300 is approximately 72.80.

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 5300 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √5300. The smallest perfect square less than 5300 is 4900, and the largest perfect square greater than 5300 is 5625. √5300 falls somewhere between 70 and 75.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square)/(Greater perfect square - smallest perfect square). Using the formula, (5300 - 4900)/(5625 - 4900) = 400/725 ≈ 0.552. 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 + 0.552 = 70.552, so the square root of 5300 is approximately 72.80.

• 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 usually the positive square root that is used, known as the principal square root.

• Prime factorization: The process of expressing a number as a product of its prime factors.

• Long division method: A step-by-step method used to find the square root of a number by dividing it into groups and finding successive approximations.