Square Root of 731

The square root is the inverse of the square of a number. 731 is not a perfect square. The square root of 731 is expressed in both radical and exponential form. In the radical form, it is expressed as √731, whereas in the exponential form it is expressed as (731)¹/². √731 ≈ 27.04163, 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:

• 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 731 is broken down into its prime factors.

Step 1: Finding the prime factors of 731 731 is a prime number itself. Hence, the prime factorization of 731 is 731.

Step 2: Since 731 is not a perfect square, calculating the square root using prime factorization directly is not feasible.

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, group the numbers from right to left. In the case of 731, group it as 31 and 7.

Step 2: Find n whose square is closest to or less than 7. Here, n is 2 because 2² = 4 is less than 7. The quotient is 2, and the remainder is 7 - 4 = 3.

Step 3: Bring down 31, making the new dividend 331. Add the old divisor (2) with itself to get 4, the start of the new divisor.

Step 4: Determine n such that 4n × n ≤ 331. Trying n = 8, we find 48 × 8 = 384, which is too large. Using n = 7, 47 × 7 = 329.

Step 5: Subtract 329 from 331 to get a remainder of 2. The quotient now is 27.

Step 6: Add a decimal point and two zeros to the dividend, making it 200.

Step 7: The new divisor is 54 (27 doubled and plus a digit to test). Continue refining until you approximate to two decimal places.

So, the square root of √731 ≈ 27.04.

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 731 using the approximation method.

Step 1: Find the closest perfect squares to √731. The smallest perfect square below 731 is 729 (27²), and the largest is 784 (28²). √731 falls between 27 and 28.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). For 731, (731 - 729) / (784 - 729) = 2 / 55 ≈ 0.036. Adding this to 27, we get 27 + 0.036 = 27.036.

So, the square root of 731 is approximately 27.04.

• Square root: A square root is the inverse operation of squaring a number. Example: If 5² = 25, then √25 = 5.

• Irrational number: An irrational number cannot be written as a simple fraction. Its decimal form goes on forever without repeating. Example: √2, √731.

• Prime number: A prime number is a number greater than 1 that has no divisors other than 1 and itself. Example: 2, 3, 5, 731.

• Approximation: Approximation refers to finding a value that is close enough to the correct answer, usually with some degree of accuracy. Example: √731 ≈ 27.04.

• Long division: A method used to divide large numbers and determine the quotient with a remainder, often used for calculating square roots of non-perfect squares.