Square Root of 853

The square root is the inverse of squaring a number. 853 is not a perfect square. The square root of 853 is expressed in both radical and exponential form. In the radical form, it is expressed as √853, whereas (853)^(1/2) in the exponential form. √853 ≈ 29.207, 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 like 853, 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 853 is broken down into its prime factors.

Step 1: Finding the prime factors of 853 853 is a prime number, and thus it cannot be broken down into smaller prime factors. Since 853 is not a perfect square, calculating √853 using prime factorization is not possible.

The long division method is particularly used for non-perfect square numbers. 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 numbers from right to left. In the case of 853, we group it as 53 and 8.

Step 2: Now, find n whose square is less than or equal to 8. We select n as 2 because 2 x 2 = 4, which is less than 8. Now the quotient is 2, and subtracting 4 from 8 leaves a remainder of 4.

Step 3: Bring down 53, making the new dividend 453. Add the old divisor with the same number: 2 + 2 = 4.

Step 4: The new divisor will be 2n. We need to find the value of n.

Step 5: The next step is finding 2n x n ≤ 453. Let us consider n as 9, now 49 x 9 = 441.

Step 6: Subtract 441 from 453, leaving a remainder of 12, 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 1200.

Step 8: Find the new divisor, which is 58, because 582 x 2 = 1164.

Step 9: Subtracting 1164 from 1200 gives a result of 36.

Step 10: The quotient is 29.2.

Step 11: Continue these steps until we achieve the desired precision.

So the square root of √853 is approximately 29.207.

The approximation method is another way to find square roots. It is an easy method to approximate the square root of a given number. Now let us learn how to find the square root of 853 using the approximation method.

Step 1: Find the closest perfect squares around √853. The smallest perfect square less than 853 is 841, and the largest perfect square greater than 853 is 900. √853 falls somewhere between 29 and 30.

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

Using the formula: (853 - 841) / (900 - 841) = 12 / 59 ≈ 0.203. Adding the initial whole number to the decimal: 29 + 0.203 = 29.203, so the square root of 853 is approximately 29.203.

• Square root: A square root is the inverse of squaring a number. 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.
   
• Prime number: A prime number is a number greater than 1 with no divisors other than 1 and itself.
   
• Decimal: If a number has a whole number and a fraction in a single number, it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.
   
• Perfect square: A perfect square is an integer that is the square of an integer. For example, 16 is a perfect square because it is 4^2.