Square Root of 859

The square root is the inverse of the square of the number. 859 is not a perfect square. The square root of 859 is expressed in both radical and exponential form. In the radical form, it is expressed as √859, whereas (859)^(1/2) in the exponential form. √859 ≈ 29.3087, 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 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 859 is broken down into its prime factors.

Step 1: Finding the prime factors of 859

Breaking it down, we get 859 = 7 × 7 × 17.

Step 2: Now we found the prime factors of 859. The second step is to make pairs of those prime factors. Since 859 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 859 using prime factorization is impossible.

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 859, we need to group it as 59 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 × 2 = 4, which is less than 8. Now the quotient is 2, and after subtracting 4 from 8, the remainder is 4.

Step 3: Now bring down 59, which is the new dividend. Add the old divisor with the same number, 2 + 2 = 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, and we need to find the value of n.

Step 5: The next step is finding 4n × n ≤ 459. Let us consider n as 9, now 49 × 9 = 441.

Step 6: Subtract 441 from 459; the difference is 18, 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 1800.

Step 8: Now we need to find the new divisor, which is 586, because 586 × 3 = 1758.

Step 9: Subtracting 1758 from 1800, we get the result 42.

Step 10: Now the quotient is 29.3

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

So the square root of √859 is approximately 29.31

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 859 using the approximation method.

Step 1: Now we have to find the closest perfect square of √859. The smallest perfect square less than 859 is 841, and the largest perfect square greater than 859 is 900. √859 falls somewhere between 29 and 30.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Going by the formula (859 - 841) / (900 - 841) = 18/59 ≈ 0.31

Using the formula, we identified the decimal point of our square root. The next step is adding the integer value to the decimal number, which is 29 + 0.31 = 29.31, so the square root of 859 is approximately 29.31.

• 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.
   
• Prime factorization: Prime factorization is expressing a number as the product of its prime factors. Example: The prime factorization of 859 is 7 × 7 × 17.
   
• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.
   
• Long division method: It is a method used to find the square root of non-perfect square numbers by dividing the number step by step to find the quotient.