Square Root of 892

The square root is the inverse of the square of the number. 892 is not a perfect square. The square root of 892 is expressed in both radical and exponential form. In the radical form, it is expressed as √892, whereas (892)^(1/2) in the exponential form. √892 ≈ 29.866, 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 892 is broken down into its prime factors:

Step 1: Finding the prime factors of 892 Breaking it down, we get 2 x 2 x 223: 2² x 223¹

Step 2: Now we found the prime factors of 892. The second step is to make pairs of those prime factors. Since 892 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 892 using prime factorization directly is not possible to find an exact integer result.

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 892, we need to group it as 92 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 from 8, the remainder is 4.

Step 3: Now, let us bring down 92 which is the new dividend. 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 the sum of the old divisor and the quotient. Now we get 4n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 4n x n ≤ 492. Let us consider n as 9, now 49 x 9 = 441.

Step 6: Subtract 441 from 492; the difference is 51, and the quotient is 29.

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 zeroes to the dividend. Now the new dividend is 5100.

Step 8: Now we need to find the new divisor that is 598 because 598 x 8 = 4784.

Step 9: Subtracting 4784 from 5100, we get the result 316. Step 10: Now the quotient is 29.8.

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

So the square root of √892 is approximately 29.87.

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 892 using the approximation method.

Step 1: Now we have to find the closest perfect square of √892.

The smallest perfect square less than 892 is 841, and the largest perfect square greater than 892 is 900.

√892 falls somewhere between 29 and 30.

Step 2: Now we need to apply the formula:

(Given number - smaller perfect square) / (Greater perfect square - smaller perfect square).

Going by the formula (892 - 841) / (900 - 841) = 51 / 59 ≈ 0.864.

Using the formula, we identified the decimal point of our square root.

The next step is adding the value we got initially to the decimal number which is 29 + 0.864 = 29.864, so the square root of 892 is approximately 29.864.

• Square root: A square root is the inverse of a square. Example: 4² = 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, it is always the positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as a principal square root.
   
• Prime factorization: The process of breaking a number down into its basic prime factors.
   
• Long division method: A technique used to find the square root of non-perfect squares by dividing and averaging.