Square Root of 73

The square root is the inverse of the square of the number. 73 is not a perfect square. The square root of 73 is expressed in both radical and exponential form.

In the radical form, it is expressed as √73, whereas (73^)(1/2) in the exponential form. √73 ≈ 8.544, 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 long-division and approximation methods are used. Let us now learn the following methods:

1. Prime factorization method
2. Long division method
3. Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 73 is broken down into its prime factors.

Step 1: Finding the prime factors of 73

73 is a prime number, so it can only be divided by 1 and itself. Therefore, the prime factorization of 73 is simply 73 itself.

Step 2: Since 73 is not a perfect square, calculating its square root using prime factorization is not feasible.

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 73, we can consider the entire number as a group.

Step 2: Now we need to find n whose square is less than or equal to 73. We can say n as ‘8’ because 8 × 8 = 64, which is less than 73. Now the quotient is 8, and after subtracting 64 from 73, the remainder is 9.

Step 3: Add a decimal point to the quotient and bring down a pair of zeros, making the new dividend 900.

Step 4: The new divisor will be 2 times the current quotient, which is 16. We need to find a digit x such that 16x × x ≤ 900. Let x be 5, then 165 × 5 = 825.

Step 5: Subtract 825 from 900, and the remainder is 75.

Step 6: Since the remainder doesn't reach zero, continue with the process by bringing down more pairs of zeros and repeating the steps to find more precise digits.

So, the square root of √73 is approximately 8.54.

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 73 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √73. The smallest perfect square less than 73 is 64, and the largest perfect square greater than 73 is 81. Therefore, √73 falls somewhere between 8 and 9.

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 (73 - 64) / (81 - 64) = 9/17 ≈ 0.53 Using the formula, we identified the decimal point of our square root.

The next step is adding this value to the lower integer value, which is 8 + 0.53 ≈ 8.53, so the square root of 73 is approximately 8.53.

• 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.

• Prime number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

• Long division method: A method used to find the square root of non-perfect squares by dividing the number into parts and approximating the root step by step.

• Approximation method: A method of estimating the square root of a number by finding two perfect squares between which the number lies and calculating a closer estimate.