Square Root of 949

The square root is the inverse of the square of the number. 949 is not a perfect square. The square root of 949 is expressed in both radical and exponential form. In the radical form, it is expressed as √949, whereas (949)^(1/2) in the exponential form. √949 ≈ 30.80584, 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 949 is broken down into its prime factors:

Step 1: Finding the prime factors of 949 Breaking it down, we get 13 x 73.

Step 2: Now we found out the prime factors of 949. The second step is to make pairs of those prime factors. Since 949 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 949 using prime factorization is not feasible for finding the 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 949, we need to group it as 49 and 9.

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 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 49, 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 6n. We need to find the value of n such that 6n x n is less than or equal to 49.

Step 5: Let n be 0, so 60 x 0 = 0.

Step 6: Subtracting 0 from 49, the difference is 49, and the quotient is 30.

Step 7: Since the dividend is greater 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 4900.

Step 8: Now we need to find the new divisor. Let n be 8 because 608 x 8 = 4864.

Step 9: Subtracting 4864 from 4900 gives the result 36. Step 10: The quotient is now 30.8.

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 √949 ≈ 30.80584.

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 949 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √949. The smallest perfect square less than 949 is 900, and the largest perfect square greater than 949 is 961. √949 falls somewhere between 30 and 31.

Step 2: Now we need to apply the formula: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square).

Applying the formula: (949 - 900) / (961 - 900) = 49 / 61 ≈ 0.8033. 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 30 + 0.8033 ≈ 30.8033.

So the square root of 949 is approximately 30.80584.

• Square root: A square root is the inverse of a square. For example: 4² = 16, and the inverse of the square is the square root, which 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 also known as the principal square root.

• Long division method: A method used to find the square root of numbers that are not perfect squares by dividing and averaging.

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