Square Root of 741

The square root is the inverse of squaring a number. 741 is not a perfect square. The square root of 741 is expressed in both radical and exponential form. In radical form, it is expressed as √741, and in exponential form as (741)^(1/2). √741 ≈ 27.213, 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, for non-perfect square numbers like 741, the 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 741 is broken down into its prime factors.

Step 1: Finding the prime factors of 741

Breaking it down, we get 3 x 13 x 19: 3^1 x 13^1 x 19^1

Step 2: Now we found the prime factors of 741. Since 741 is not a perfect square, the digits of the number can’t be grouped in pairs that result in a whole number. Therefore, calculating 741 using prime factorization alone is insufficient for finding its 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 741, we need to group it as 41 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, which is less than 7. The quotient is 2, and after subtracting 4 from 7, the remainder is 3.

Step 3: Bring down the next pair, which is 41, to make the new dividend 341. Add the old divisor, which was 2, to itself, getting 4 as the new tentative divisor.

Step 4: Determine the next digit of the quotient by finding n such that 4n x n ≤ 341. Choosing n as 8, we get 48 x 8 = 384, which is too large. Trying n as 7, we get 47 x 7 = 329, which fits.

Step 5: Subtract 329 from 341 to get a remainder of 12.

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, making it 1200.

Step 7: Repeat the process to get more decimal places. The next digit of the quotient is determined using the divisor 547. Continuing this process gives us the square root of 741 as approximately 27.213.

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 741 using the approximation method.

Step 1: Identify the closest perfect squares around 741. The smallest perfect square less than 741 is 729 (27^2), and the largest perfect square greater than 741 is 784 (28^2). So, √741 falls between 27 and 28.

Step 2: Use the approximation formula: (Given number - smaller perfect square) ÷ (larger perfect square - smaller perfect square). Using the formula: (741 - 729) ÷ (784 - 729) = 12 ÷ 55 ≈ 0.218 Adding the approximate value to the lower bound gives: 27 + 0.218 ≈ 27.218. Therefore, the square root of 741 is approximately 27.218.

• Square Root: A square root is the inverse of squaring a number. Example: 4^2 = 16, and the inverse is the square root: √16 = 4.

• Irrational Number: An irrational number cannot be expressed as a fraction p/q, where p and q are integers, and q ≠ 0.

• Principal Square Root: A number has both positive and negative square roots, but the positive square root is typically used in practical applications. This is known as the principal square root.

• Decimal: A decimal is a number that includes a whole number and a fractional part, such as 7.86, 8.65, and 9.42.

• Long Division Method: A process used to find the square root of non-perfect squares by dividing the number into parts and estimating the quotient step by step.