Square Root of 969

The square root is the inverse of the square of a number. 969 is not a perfect square. The square root of 969 is expressed in both radical and exponential form. In the radical form, it is expressed as √969, whereas in exponential form as (969)^(1/2). √969 ≈ 31.1136, 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 typically used for non-perfect square numbers, where the long division method and approximation method are more suitable. 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 969 is broken down into its prime factors.

Step 1: Finding the prime factors of 969 Breaking it down, we get 3 x 17 x 19.

Step 2: Now we have found the prime factors of 969. The second step is to make pairs of those prime factors. Since 969 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 969 using prime factorization does not yield a simplified square root.

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 969, we need to group it as 69 and 9.

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

Step 3: Now let us bring down 69, which is the new dividend. Add the old divisor with the same number 3 + 3 to get 6, which will be our new divisor.

Step 4: 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 ≤ 69. Let us consider n as 1, now 6 x 1 x 1 = 6.

Step 6: Subtract 6 from 69, the difference is 63, and the quotient becomes 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 6300.

Step 8: Now we need to find the new divisor, which is 621 because 621 x 1 = 621.

Step 9: Subtracting 621 from 6300 gives the result 79.

Step 10: Now the quotient is 31.1.

Step 11: Continue doing these steps until we get two numbers after the decimal point.

Suppose if there is no decimal value, continue until the remainder is zero. So the square root of √969 ≈ 31.11.

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 969 using the approximation method.

Step 1: Now we have to find the closest perfect squares around √969. The smallest perfect square less than 969 is 961 (31²), and the largest perfect square greater than 969 is 1024 (32²). √969 falls 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: (969 - 961) / (1024 - 961) ≈ 0.13.

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 + 0.13 = 31.13.

So the square root of 969 is approximately 31.13.

• 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, but the positive square root, known as the principal square root, is often used due to its relevance in real-world applications.

• Long division method: A step-by-step approach used to find the square root of a non-perfect square number by dividing the number into groups of two digits.

• Decimal: A number that includes a whole number and a fraction represented with a decimal point, such as 7.86, 8.65, and 9.42.