Square Root of 7321

The square root is the inverse of the square of a number. 7321 is not a perfect square. The square root of 7321 is expressed in both radical and exponential form: √7321 in radical form and (7321)^(1/2) in exponential form. √7321 ≈ 85.5561, 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, 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. However, since 7321 is not a perfect square, prime factorization is not suitable for finding its square root. Instead, the long division or approximation method is more appropriate.

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 7321, we start with 73 and 21.

Step 2: Now we need to find n whose square is less than or equal to 73. Here, n is 8 because 8 x 8 = 64, which is less than 73. The quotient is 8, and the remainder is 73 - 64 = 9.

Step 3: Bring down the next pair of digits, 21, making the new dividend 921.

Step 4: Double the quotient and write it with a blank space (16_), then find a digit to fill the blank that, when multiplied by the result, gives a product less than or equal to 921.

Step 5: Complete the division process, continuing to bring down pairs of zeros as necessary, to find more decimal places in the square root. Following this procedure, we find that √7321 ≈ 85.5561.

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 7321 using the approximation method.

Step 1: Identify the closest perfect squares around 7321. The smallest perfect square less than 7321 is 7225, and the largest perfect square greater than 7321 is 7396. Thus, √7321 falls between 85 and 86.

Step 2: Use linear interpolation or estimation between these numbers to find a more exact value. Using this method, we approximate √7321 ≈ 85.5561.

• Square root: A square root is the inverse operation of squaring a number. Example: 4² = 16, and the inverse square root is √16 = 4.
   
• Irrational number: An irrational number cannot be expressed as a ratio of two integers. Its decimal form is non-repeating and non-terminating.
   
• Long division method: A manual arithmetic technique to find the square root of numbers, particularly non-perfect squares, by a series of division steps.
   
• Approximation method: A technique to estimate the square root of a number by identifying nearby perfect squares and interpolating between them
   
• Factors: Numbers that divide another number exactly without leaving a remainder. For example, factors of 10 are 1, 2, 5, and 10.