Square Root of 740

The square root is the inverse of the square of the number. 740 is not a perfect square. The square root of 740 is expressed in both radical and exponential form. In the radical form, it is expressed as √740, whereas (740)^(1/2) in the exponential form. √740 ≈ 27.1928, 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:

• Prime factorization method  
• Long division method 
• Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 740 is broken down into its prime factors:

Step 1: Finding the prime factors of 740

Breaking it down, we get 2 x 2 x 5 x 37: 2^2 x 5 x 37

Step 2: Now we found out the prime factors of 740. The second step is to make pairs of those prime factors. Since 740 is not a perfect square, therefore the digits of the number can’t be grouped in pairs. Therefore, calculating 740 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 740, we need to group it as 40 and 7.

Step 2: Now we need to find n whose square is less than or equal to 7. We can say n is ‘2’ because 2 x 2 = 4 is less than 7. Now the quotient is 2, and after subtracting 4 from 7, the remainder is 3.

Step 3: Now let us bring down 40, which is the new dividend. Add the old divisor with the same number 2 + 2 to get 4, which will be our new divisor.

Step 4: The new divisor will be 4n. We need to find the value of n such that 4n x n ≤ 340. Let us consider n as 8; now 48 x 8 = 384, which is too high. Trying n as 7, we get 47 x 7 = 329, which fits.

Step 5: Subtract 329 from 340, the difference is 11, and the quotient is 27.

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. Now the new dividend is 1100.

Step 7: The new divisor becomes 547. We determine that n is 2 because 547 x 2 = 1094.

Step 8: Subtracting 1094 from 1100, we get the result 6.

Step 9: Now the quotient is 27.2

Step 10: Continue doing these steps until we get two numbers after the decimal point or until the remainder is zero.

So the square root of √740 is approximately 27.1928.

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 740 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √740. The smallest perfect square less than 740 is 729, and the largest perfect square greater than 740 is 784. √740 falls somewhere between 27 and 28.

Step 2: Now we need to apply the formula (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Using this, (740 - 729) ÷ (784 - 729) = 11 ÷ 55 ≈ 0.2 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 27 + 0.2 = 27.2, so the square root of 740 is approximately 27.2

• Square root: A square root is the inverse of a square. Example: 4^2 = 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.

• Perfect square: A perfect square is a number that is the square of an integer. Example: 36 is a perfect square because it is 6^2.

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

• Prime factorization: Prime factorization is expressing a number as the product of its prime factors. Example: The prime factorization of 28 is 2 x 2 x 7. ```