Square Root of 857

The square root is the inverse of the square of the number. 857 is not a perfect square. The square root of 857 is expressed in both radical and exponential form. In radical form, it is expressed as √857, whereas (857)^(1/2) in exponential form. √857 ≈ 29.273, 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, 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 857 is broken down into its prime factors.

Step 1: Finding the prime factors of 857. 857 is a prime number, so it cannot be broken down into other prime factors.

Step 2: Since 857 is not a perfect square, calculating √857 using prime factorization is not possible.

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 group the numbers from right to left. In the case of 857, we need to group it as 57 and 8.

Step 2: Now find n whose square is ≤ 8. We can say n is 2 because 2 × 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 57, the new dividend becomes 457. Add the old divisor with the same number 2 + 2 = 4, which will be our new divisor prefix.

Step 4: The new divisor is 4n. Find n such that 4n × n is ≤ 457. Let n be 9, thus 49 × 9 = 441.

Step 5: Subtract 441 from 457; the difference is 16, and the quotient is 29.

Step 6: Since the dividend is less than the divisor, we add a decimal point and bring down two zeroes, making the new dividend 1600.

Step 7: Find the new divisor 58n, where n is such that 58n × n ≤ 1600. Try n = 2, giving 582 × 2 = 1164.

Step 8: Subtract 1164 from 1600 to get 436.

Step 9: Continue this process until the desired precision is achieved.

So the square root of √857 ≈ 29.273.

The approximation method is another way to find square roots. It is an easy method to find the square root of a given number. Let us learn how to find the square root of 857 using the approximation method.

Step 1: Find the closest perfect squares to √857.

The smallest perfect square less than 857 is 841 (29^2) and the largest is 900 (30^2). So √857 is between 29 and 30.

Step 2: Use the formula: (Given number - smallest perfect square) / (Larger perfect square - smallest perfect square). (857 - 841) / (900 - 841) = 16 / 59 ≈ 0.271.

Step 3: Add this decimal to the smaller integer: 29 + 0.271 = 29.271.

So the square root of 857 is approximately 29.273.

• Square root: The square root is the inverse operation of squaring a number. For example, 5^2 = 25, and the inverse is √25 = 5.
   
• Prime number: A prime number is a natural number greater than 1 that is not divisible by any other numbers except for 1 and itself.
   
• Irrational number: An irrational number cannot be expressed as a simple fraction or ratio of two integers.
   
• Long division method: A method used to find the square root of non-perfect squares by dividing the number into groups and performing iterative division steps.
   
• Radical: The symbol (√) used to denote the square root or nth root of a number.