Square Root of 170000

The square root is the inverse of the square of a number. 170000 is not a perfect square. The square root of 170000 can be expressed in both radical and exponential forms. In radical form, it is expressed as √170000, whereas (170000)^(1/2) in exponential form. √170000 ≈ 412.3106, 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 and approximation methods 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 170000 is broken down into its prime factors.

Step 1: Finding the prime factors of 170000. Breaking it down, we get 2 x 2 x 2 x 5 x 5 x 5 x 17 x 17: 2^3 x 5^3 x 17^2.

Step 2: Now we found out the prime factors of 170000. The second step is to make pairs of those prime factors. Since 170000 is not a perfect square, the digits of the number can’t be grouped into complete pairs. Therefore, calculating 170000 using prime factorization requires further steps, leading to an approximate square root.

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 170000, we can group it as 00, 00, and 17.

Step 2: Now we need to find n whose square is less than or equal to 17. We can say n is 4 because 4 x 4 = 16. Now the quotient is 4, and after subtracting 16 from 17, the remainder is 1.

Step 3: Bring down the next pair of digits (00) to get 100. Add 4 to the old divisor (4) to get 8, which will be our new divisor (80).

Step 4: Find the largest number x such that 80x x x ≤ 100. The largest such x is 1, since 801 x 1 = 80.

Step 5: Subtract 80 from 100 to get 20. Bring down the next pair of zeros (00) to get 2000.

Step 6: Continue the process until you achieve the desired decimal places. So the approximate square root of √170000 is 412.3106.

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 170000 using the approximation method.

Step 1: Identify the closest perfect squares around 170000. The smallest perfect square less than 170000 is 169000 (410^2), and the largest perfect square greater than 170000 is 171396 (414^2).

Step 2: Use the approximation formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula: (170000 - 169000) / (171396 - 169000) = 1000 / (2396) ≈ 0.4173. Adding the approximate decimal to the closest smaller whole number, we get 410 + 0.4173 ≈ 412.3106.

• 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.

• Principal square root: A number has both positive and negative square roots; however, it is the positive square root that is typically used in real-world applications. This is known as the principal square root.

• Prime factorization: The expression of a number as the product of its prime factors. Example: The prime factorization of 100 is 2^2 x 5^2.

• Long division method: A method used to find the square root of a number, especially non-perfect squares, by using division and subtraction iteratively.