Square Root of 882

The square root is the inverse of the square of the number. 882 is not a perfect square. The square root of 882 is expressed in both radical and exponential form. In the radical form, it is expressed as √882, whereas (882)^(1/2) in the exponential form. √882 ≈ 29.69848, 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 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 882 is broken down into its prime factors:

Step 1: Finding the prime factors of 882

Breaking it down, we get 2 x 3 x 3 x 7 x 7: 2^1 x 3^2 x 7^2

Step 2: Now we found the prime factors of 882. The second step is to make pairs of those prime factors. Since 882 is not a perfect square, the digits cannot be grouped perfectly in pairs. Thus, calculating the square root of 882 using prime factorization alone is challenging.

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 882, we need to group it as 82 and 8.

Step 2: Now we need to find n whose square is 8. We can say n as ‘2’ because 2 x 2 is lesser than or equal to 8. Now the quotient is 2, and after subtracting 4 (2x2) from 8, the remainder is 4.

Step 3: Now let us bring down 82, making the new dividend 482. Add the old divisor with the same number, 2 + 2, to get 4, which will be our new divisor.

Step 4: The new divisor will be 4n. We need to find the value of n.

Step 5: Find 4n x n ≤ 482. Let n be 7, then 47 x 7 = 329.

Step 6: Subtract 329 from 482, and the difference is 153.

Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeros to the dividend. Now the new dividend is 15300.

Step 8: Find the new divisor, which is 594, because 594 x 2 = 1188.

Step 9: Subtracting 1188 from 15300 gives 3412.

Step 10: The quotient is 29.6.

Step 11: Continue doing these steps until we get two numbers after the decimal point. If there is no decimal value, continue till the remainder is zero.

So the square root of √882 is approximately 29.70.

The approximation method is another way to find square roots and is an easy method to find the square root of a given number. Let us learn how to find the square root of 882 using the approximation method.

Step 1: Find the closest perfect squares of √882. The smallest perfect square less than 882 is 841, and the largest perfect square greater than 882 is 900. √882 falls somewhere between 29 and 30.

Step 2: Apply the formula (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square)

Using the formula (882 - 841) / (900 - 841) = 41/59 ≈ 0.69 Adding this value to the smaller perfect square's root: 29 + 0.69 = 29.69, so the square root of 882 is approximately 29.69.

• Square root: A square root is the inverse of a square. Example: 4^2 = 16, and the inverse of the square is the square root, that is √16 = 4.
   
• Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.
   
• Principal square root: A number has both positive and negative square roots; however, the positive square root is more commonly used due to its real-world applications. This is known as the principal square root.
   
• Prime factorization: Breaking down a number into a product of its prime factors.
   
• Long division method: A method used to find the square root of non-perfect squares through systematic division.