Square Root of 940

The square root is the inverse of the square of a number. 940 is not a perfect square. The square root of 940 is expressed in both radical and exponential form. In radical form, it is expressed as √940, whereas (940)^(1/2) in exponential form. √940 ≈ 30.662, 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 often used for perfect square numbers. However, for non-perfect square numbers, 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 940 is broken down into its prime factors:

Step 1: Finding the prime factors of 940 Breaking it down, we get 2 x 2 x 5 x 47: 2^2 x 5 x 47

Step 2: We found the prime factors of 940. The second step is to make pairs of those prime factors. Since 940 is not a perfect square, the digits of the number can’t be grouped in pairs perfectly.

Therefore, calculating 940 using prime factorization directly is not feasible for finding its exact 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 940, we need to group it as 40 and 9.

Step 2: Now, we need to find n whose square is 9. We can say n is ‘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 40, 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: The new divisor will be 6n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 6n x n ≤ 40. Let us consider n as 6, now 6 x 6 = 36.

Step 6: Subtract 40 from 36, and the difference is 4, and the quotient is 36.

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 400.

Step 8: Now we need to find the new divisor that’s 61 because 61 x 6 = 366.

Step 9: Subtracting 366 from 400, we get the result 34.

Step 10: Now the quotient is 30.6.

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 √940 is approximately 30.66.

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 940 using the approximation method.

Step 1: Now we have to find the closest perfect square to √940. The smallest perfect square less than 940 is 900, and the largest perfect square more than 940 is 961. √940 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).

Using the formula: (940 - 900) / (961 - 900) = 40 / 61 ≈ 0.6557. Adding this to 30, we get approximately 30.66.

So, the square root of 940 is approximately 30.66.

• 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, the positive square root is often used in real-world applications and is known as the principal square root.

• Prime factorization: Expressing a number as the product of its prime factors. For example, the prime factorization of 940 is 2² x 5 x 47.

• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example, 7.86, 8.65, and 9.42 are decimals.