Square Root of 93

The square root is the inverse of the square of the number. 93 is not a perfect square. The square root of 93 is expressed in both radical and exponential forms.

In the radical form, it is expressed as √93, whereas (93)^(1/2) in the exponential form. √93 ≈ 9.64365, 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 method and approximation method 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 93 is broken down into its prime factors.

Step 1: Finding the prime factors of 93 Breaking it down, we get 3 x 31: 3¹ x 31¹

Step 2: Now we found out the prime factors of 93. The second step is to make pairs of those prime factors. Since 93 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 93 using prime factorization is impossible for perfect pairing.

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 93, we need to group it as 93.

Step 2: Now we need to find a number whose square is close to or less than 93. We can say it is '9' because 9² is 81, which is lesser than or equal to 93. Now the quotient is 9, and after subtracting 81 from 93, the remainder is 12.

Step 3: Since the remainder is not zero, we need to add a decimal point and bring down two zeros to make it 1200.

Step 4: The new divisor is 2 times the quotient from step 2, which is 18. We need to find a digit 'd' such that 18d × d is less than or equal to 1200.

Step 5: By trial, we find that 186 × 6 = 1116, which is less than 1200. Subtract 1116 from 1200 to get 84.

Step 6: Bring down two more zeros to make it 8400, and repeat the process: 192d × d, finding the digit 'd' that works.

Step 7: Continue doing these steps until we get a satisfactory approximation. After enough iterations, the quotient will approach 9.64.

So the square root of √93 is approximately 9.64.

The approximation method is another method for finding the 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 93 using the approximation method.

Step 1: Now we have to find the closest perfect square of √93. The smallest perfect square less than 93 is 81, and the largest perfect square greater than 93 is 100. √93 falls somewhere between 9 and 10.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Applying the formula (93 - 81) ÷ (100 - 81) = 12 ÷ 19 ≈ 0.63.

Using this approximation, we adjust from the smaller perfect square root: 9 + 0.63 ≈ 9.63, so the square root of 93 is approximately 9.63.

• 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 a principal square root.

• Prime factorization: The process of breaking down a number into its prime factors. For example, the prime factorization of 93 is 3 x 31.

• Long division method: A method used to find the square root of non-perfect squares by using repeated division.