Square Root of 250000

The square root is the inverse of the square of the number. 250000 is a perfect square. The square root of 250000 is expressed in both radical and exponential form. In the radical form, it is expressed as √250000, whereas (250000)^(1/2) in the exponential form. √250000 = 500, which is a rational number because it can 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. For perfect square numbers like 250000, this method can be efficiently 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 250000 is broken down into its prime factors.

Step 1: Finding the prime factors of 250000 Breaking it down, we get 2 x 2 x 2 x 2 x 5 x 5 x 5 x 5: 2^4 x 5^4

Step 2: Now we found out the prime factors of 250000. The second step is to make pairs of those prime factors. Since 250000 is a perfect square, the digits of the number can be grouped in pairs. Therefore, calculating √250000 using prime factorization gives us 500.

The long division method is particularly used for non-perfect square numbers, but it can also be applied to perfect squares to verify results. 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 250000, we need to group it as 00, 50, and 25.

Step 2: Now we need to find n whose square is 25. We can say n as ‘5’ because 5 x 5 is equal to 25. Now the quotient is 5 after subtracting 25 - 25, the remainder is 0.

Step 3: Now bring down 00, making the new dividend 000.

Step 4: Add the old divisor with the same number 5 + 5 to get 10, which will be our new divisor.

Step 5: The next step is to find 10n x n ≤ 000. Since the dividend is 000, we bring down the next pair of numbers, 00, and proceed accordingly.

Step 6: Continue this process until all numbers are divided, resulting in the square root being 500.

The approximation method is useful for estimating square roots. However, for a perfect square, the exact value is already known. Let us consider how it might be done for 250000.

Step 1: Identify the closest perfect square numbers. Since 250000 is a perfect square, the square root is exactly 500.

Students do make mistakes while finding the square root, such as overlooking the simplicity of perfect squares. Let us look at a few of the common mistakes.

• Square root: A square root is the inverse of a square. Example: 20^2 = 400, and the inverse of the square is the square root that is √400 = 20.

• Rational number: A rational number is a number that can be written in the form of p/q, where q is not equal to zero and p and q are integers.

• Perfect square: A perfect square is a number that is the square of an integer. 250000 is a perfect square because 500 x 500 = 250000.

• Integer: An integer is a whole number that can be positive, negative, or zero. Examples include -3, 0, 7.

• Factor: A factor is a number that divides another number without leaving a remainder. For example, 5 is a factor of 25.