Square Root of 977

The square root is the inverse of the square of a number. 977 is not a perfect square. The square root of 977 is expressed in both radical and exponential form. In the radical form, it is expressed as √977, whereas (977)^(1/2) is the exponential form. √977 ≈ 31.2569992, 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 977 is broken down into its prime factors.

Step 1: Finding the prime factors of 977 977 is a prime number itself. Hence, it cannot be broken down into smaller prime factors.

Since 977 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 977, we need to group it as 77 and 9.

Step 2: Now we need to find n whose square is ≤ 9. We can say n is ‘3’ because 3 x 3 = 9. Now the quotient is 3 and the remainder is 0 after subtracting 9 - 9.

Step 3: Now let us bring down 77 which is the new dividend. Add the old divisor with the same number 3 + 3 = 6 which will be our new divisor.

Step 4: The new divisor will be 6x. Now we need to find a digit x such that 6x x x ≤ 77. Trying x = 1, 61 x 1 = 61.

Step 5: Subtract 61 from 77, the remainder is 16.

Step 6: 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. The new dividend is 1600.

Step 7: The new divisor is 62 because 622 x 2 = 1244.

Step 8: Subtracting 1244 from 1600, we get the result 356. Step 9: Now the quotient is 31.2

Step 10: Continue doing these steps until we get the desired precision. Suppose if there are no decimal values, continue till the remainder is zero.

So the square root of √977 is approximately 31.256.

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 977 using the approximation method.

Step 1: Now we have to find the closest perfect square of √977. The smallest perfect square less than 977 is 961 (31 x 31), and the largest perfect square greater than 977 is 1024 (32 x 32). Therefore, √977 falls somewhere between 31 and 32.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square)

Using the formula (977 - 961) ÷ (1024 - 961) = 16 ÷ 63 ≈ 0.254 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 + 0.254 ≈ 31.254, so the square root of 977 is approximately 31.254.

• 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. This is why it is also known as the principal square root.

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

• Perfect square: A perfect square is a number that is the square of an integer. For example, 9 is a perfect square because it is 3².