Square Root of 976

The square root is the inverse of the square of a number. 976 is not a perfect square. The square root of 976 is expressed in both radical and exponential form. In the radical form, it is expressed as √976, whereas (976)^(1/2) in the exponential form. √976 ≈ 31.24099, 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 976 is broken down into its prime factors.

Step 1: Finding the prime factors of 976 Breaking it down, we get 2 x 2 x 2 x 2 x 61: 2⁴ x 61¹

Step 2: Now we found out the prime factors of 976. The second step is to make pairs of those prime factors. Since 976 is not a perfect square, therefore, the digits of the number can’t be grouped in pairs.

Therefore, calculating 976 using prime factorization is impossible.

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 976, we need to group it as 76 and 9.

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

Step 3: Now let us bring down 76, which is the new dividend. Add the old divisor with the same number 3 + 3, we get 6, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 6n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 6n x n ≤ 76. Let us consider n as 1, now 6 x 1 x 1 = 61.

Step 6: Subtract 76 from 61; the difference is 15, 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 1500.

Step 8: Now we need to find the new divisor that is 312 because 312 x 4 = 1248.

Step 9: Subtracting 1248 from 1500, we get the result 252.

Step 10: Now the quotient is 31.24.

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

So the square root of √976 is approximately 31.24.

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 976 using the approximation method.

Step 1: Now we have to find the closest perfect square of √976. The closest perfect square less than 976 is 961 (which is 31²), and the closest perfect square greater than 976 is 1024 (which is 32²). So, √976 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 (976 - 961) ÷ (1024 - 961) ≈ 0.24. 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 + 0.24 = 31.24.

So the square root of 976 is approximately 31.24.

• 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.

• Factors: Factors are integers that can be multiplied together to get another integer. For example, 1, 2, 4 are factors of 8.

• Perimeter: The perimeter is the distance around a two-dimensional shape, such as a rectangle or a square.