Square Root of 852

The square root is the inverse of the square of the number. 852 is not a perfect square. The square root of 852 is expressed in both radical and exponential form. In the radical form, it is expressed as √852, whereas (852)^(1/2) in the exponential form. √852 ≈ 29.189, 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 852 is broken down into its prime factors.

Step 1: Finding the prime factors of 852.

Breaking it down, we get 2 x 2 x 3 x 71: 2^2 x 3^1 x 71^1

Step 2: Now we have found the prime factors of 852. The second step is to make pairs of those prime factors. Since 852 is not a perfect square, therefore the digits of the number can’t be grouped in pairs. Therefore, calculating 852 using prime factorization is not straightforward.

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 852, we need to group it as 52 and 8.

Step 2: Now we need to find n whose square is less than or equal to 8. We can say n as '2' because 2 x 2 = 4 is lesser than or equal to 8. Now the quotient is 2, after subtracting 4 from 8 the remainder is 4.

Step 3: Now let us bring down 52 which is the new dividend. Add the old divisor with the same number, 2 + 2, we get 4 which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 4n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 4n × n ≤ 452. Let us consider n as 1, now 41 x 1 = 41, which is less than 52.

Step 6: Subtract 41 from 52, the difference is 11, and the quotient becomes 21.

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 1100.

Step 8: Now we need to find the new divisor, which is 421.

Step 9: Subtracting 421 from 1100 gives us a new remainder, and continuing these steps will eventually give us a square root approximation.

So the square root of √852 ≈ 29.189

The approximation method is another method for finding the 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 852 using the approximation method.

Step 1: Now we have to find the closest perfect square of √852. The smallest perfect square less than 852 is 841, and the largest perfect square greater than 852 is 900. √852 falls somewhere between 29 and 30.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (852 - 841) / (900 - 841) = 11/59 ≈ 0.186

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 + 0.186 ≈ 29.186, so the square root of 852 is approximately 29.186.

• 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, 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 why it is also known as the principal square root.
   
• Prime factorization: The process of breaking down a number into its prime factors, which are the prime numbers that multiply together to give the original number.
   
• Long division method: A mathematical procedure used to find the square root of a non-perfect square number by dividing the number into smaller, more manageable parts.