Square Root of 964

The square root is the inverse of the square of the number. 964 is not a perfect square. The square root of 964 is expressed in both radical and exponential form. In the radical form, it is expressed as √964, whereas (964)^(1/2) in the exponential form. √964 ≈ 31.0494, 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 964 is broken down into its prime factors.

Step 1: Finding the prime factors of 964 Breaking it down, we get 2 × 2 × 241: 2^2 × 241

Step 2: Now we found out the prime factors of 964. The second step is to make pairs of those prime factors. Since 964 is not a perfect square, the digits of the number can’t be grouped into pairs.

Therefore, calculating 964 using prime factorization is not straightforward.

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 964, we need to group it as 64 and 9.

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

Step 3: Now let us bring down 64, which is the new dividend. Add the old divisor (3) with the same number to get 6, which will be our new divisor.

Step 4: The new divisor will be 6n. We need to find the value of n.

Step 5: The next step is finding 6n × n ≤ 64. Let us consider n as 1; now 6 × 1 × 1 = 6.

Step 6: Subtract 64 from 6, the difference is 58, and the quotient is 31.

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 5800.

Step 8: Now we need to find the new divisor. We find it by adding 31 to 6, which is 316. Then we find n such that 316n × n ≤ 5800. Let n = 1; 316 × 1 = 316.

Step 9: Subtracting 316 from 5800, we get the result 5484.

Step 10: Now the quotient is 31.0.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal value, continue till the remainder is zero.

So the square root of √964 is approximately 31.05.

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 964 using the approximation method.

Step 1: Now we have to find the closest perfect square of √964. The smallest perfect square of 964 is 961, and the largest perfect square of 964 is 1024. √964 falls somewhere between 31 and 32.

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 (964 - 961) ÷ (1024 - 961) = 3/63 ≈ 0.048.

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 31 + 0.048 = 31.048, so the square root of 964 is approximately 31.048.

• 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 why it is known as the principal square root.

• Prime factorization: The process of breaking down a number into the product of its prime factors. For example, the prime factorization of 28 is 2 × 2 × 7.

• Long division method: A technique used to find the square root of non-perfect square numbers by dividing the number into groups and calculating step by step.