Square Root of 968

The square root is the inverse of the square of the number. 968 is not a perfect square. The square root of 968 is expressed in both radical and exponential forms. In the radical form, it is expressed as √968, whereas (968)^(1/2) in the exponential form. √968 ≈ 31.1, 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:

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 968 is broken down into its prime factors.

Step 1: Finding the prime factors of 968 Breaking it down, we get 2 × 2 × 2 × 11 × 11: 2³ × 11²

Step 2: Now that we have found the prime factors of 968, the second step is to make pairs of those prime factors. Since 968 is not a perfect square, therefore the digits of the number can’t be grouped in complete pairs.

Therefore, calculating the exact square root of 968 using prime factorization alone 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 968, we can consider it as 9 and 68.

Step 2: Now we need to find n whose square is less than or equal to 9. We can say n is ‘3’ because 3 × 3 is 9. Now the quotient is 3, and the remainder is 0.

Step 3: Now bring down 68, which is the new dividend. Add the old divisor (3) with the same number to get 6, which will be part of our new divisor.

Step 4: We double the quotient (3) and consider the next digit we need. The new divisor will be 6n.

Step 5: The next step is finding 6n × n ≤ 68; let us consider n as 1, now 6 × 1 × 1 = 6. Step 6: Subtract 68 from 6; the difference is 62.

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 6200.

Step 8: Now we need to find the new digit n for the divisor 61n. Choose n = 1.

Step 9: Subtract the product from the dividend and continue the process until you obtain a precise value.

So, the square root of √968 is approximately 31.1.

The approximation method is another method for finding the 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 968 using the approximation method.

Step 1: Now we have to find the closest perfect squares around √968. The closest perfect squares to 968 are 961 (31²) and 1024 (32²). √968 falls somewhere between 31 and 32.

Step 2: Now we need to apply a formula for approximation. Using the approximation formula, we can determine a more accurate value between the two perfect squares.

Using interpolation or an approximation formula, we find that √968 is approximately 31.1.

• Square root: A square root is the inverse operation of squaring a number. Example: 4² = 16, and the inverse of squaring is the square root, so √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.

• Long division method: A method used to find the square root of non-perfect squares, involving division and subtraction in repeated steps to approximate the root.

• Perfect square: A number that is the square of an integer. Example: 49 is a perfect square because it is 7².

• Approximation method: A technique for estimating the value of a square root by finding two consecutive perfect squares and interpolating between them.