Square Root of 866

The square root is the inverse of the square of a number. 866 is not a perfect square. The square root of 866 is expressed in both radical and exponential form. In the radical form, it is expressed as √866, whereas in exponential form it is (866)^(1/2). √866 ≈ 29.4279, 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, methods like 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 866 is broken down into its prime factors.

Step 1: Finding the prime factors of 866

Breaking it down, we get 2 x 433, where 433 is a prime number.

Step 2: We found the prime factors of 866. Since 866 is not a perfect square, the digits of the number can’t be grouped into pairs. Therefore, calculating 866 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, group the digits from right to left. In the case of 866, treat it as 86 and 6.

Step 2: Find the largest integer n such that n² is less than or equal to 86. The integer is 9 because 9² = 81.

Step 3: Subtract 81 from 86, leaving a remainder of 5. Bring down the next pair of digits (66), making the new dividend 566.

Step 4: Double the divisor (9) to get 18. Now, find a digit x such that 18x × x is less than or equal to 566. The number is 3 because 183 × 3 = 549.

Step 5: Subtract 549 from 566, leaving a remainder of 17.

Step 6: Since the remainder is less than the divisor, add a decimal point and bring down two zeros, making the new dividend 1700.

Step 7: Double the previous divisor (183) and append an appropriate digit to find the next part of the quotient. Continue this process to get more decimal places.

Following these steps, the square root of 866 is approximately 29.4279.

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 866 using the approximation method.

Step 1: Identify the closest perfect squares to 866. The smallest perfect square is 841 (29²) and the largest is 900 (30²). Hence, √866 is between 29 and 30.

Step 2: Use the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Applying the formula: (866 - 841) / (900 - 841) = 25 / 59 ≈ 0.4237 Adding this to the smaller perfect square root: 29 + 0.4237 = 29.4237 Thus, the square root of 866 is approximately 29.4237.

• Square root: A square root is the inverse operation of squaring a number. Example: 4² = 16 and the inverse is the square root, √16 = 4.
   
• Irrational number: An irrational number cannot be expressed as a simple fraction. Its decimal representation is non-repeating and non-terminating.
   
• Principal square root: The non-negative root of a number. For example, the principal square root of 25 is 5.
   
• Prime number: A number that has no divisors other than 1 and itself. For example, 433 is a prime number.
   
• Long division method: A technique used to divide numbers and can be adapted to find square roots of non-perfect squares.