Square Root of 868

The square root is the inverse operation of squaring a number. 868 is not a perfect square. The square root of 868 can be expressed in both radical and exponential forms. In radical form, it is expressed as √868, whereas in exponential form, it is expressed as (868)^(1/2). √868 ≈ 29.447, 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 long division and approximation are used. Let us explore these methods:

• Prime factorization method
• Long division method
• Approximation method

The product of prime factors is the prime factorization of a number. Let us see how 868 can be broken down into its prime factors:

Step 1: Finding the prime factors of 868

Breaking it down, we get 2 x 2 x 7 x 31: 2^2 x 7 x 31

Step 2: We found the prime factors of 868. Since 868 is not a perfect square, the digits cannot be grouped into pairs. Therefore, calculating 868 using prime factorization is not feasible for finding an exact square root.

The long division method is particularly useful for non-perfect square numbers. Let us learn how to find the square root using this method, step by step:

Step 1: Group the numbers from right to left. For 868, group it as 68 and 8.

Step 2: Find n whose square is less than or equal to 8. Here, n is 2 because 2^2 = 4, which is less than 8. The quotient is 2, and the remainder is 4.

Step 3: Bring down 68, making the new dividend 468. Add the old divisor with the same number: 2 + 2 = 4, making 4 the new divisor.

Step 4: Find n such that 4n x n ≤ 468. Suppose n is 11, then 411 x 11 = 451.

Step 5: Subtract 451 from 468, the remainder is 17, and the quotient so far is 21.

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

Step 7: Find a new divisor of 221, because 2217 x 7 = 15519.

Step 8: Subtract 15519 from 17000 to get 1481.

Step 9: The quotient is 29.4.

Step 10: Continue these steps until you have the desired decimal places.

The approximate square root of √868 is 29.45.

The approximation method is another way to find square roots. It's a relatively easy method for estimating the square root of a given number. Let's learn how to find the square root of 868 using this method.

Step 1: Find the closest perfect squares to √868. The nearest perfect square less than 868 is 841, and the nearest perfect square greater than 868 is 900. √868 falls between 29 and 30.

Step 2: Apply the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square).

Using the formula: (868 - 841) / (900 - 841) = 27/59 ≈ 0.46 Adding this to the smaller perfect square root: 29 + 0.46 = 29.46, so the square root of 868 is approximately 29.46.

• Square root: The square root is the inverse operation of squaring a number. Example: 4^2 = 16, and the inverse is √16 = 4.
   
• Irrational number: An irrational number cannot be expressed as a simple fraction, meaning it cannot be written as p/q, where p and q are integers and q ≠ 0.
   
• Perfect square: A perfect square is a number that is the square of an integer. For example, 16 is a perfect square because 4^2 = 16.
   
• Long division method: A step-by-step process for finding the square root of a number, especially useful for non-perfect squares.
   
• Approximation: Estimating the value of a number or result, often used when exact calculations are difficult or unnecessary.