Square Root of 25000

The square root is the inverse of the square of the number. 25000 is a perfect square. The square root of 25000 is expressed in both radical and exponential forms. In radical form, it is expressed as √25000, whereas (25000)^(1/2) in exponential form. √25000 = 158.113883, 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 can be used for perfect square numbers. For non-perfect squares, the long-division and approximation methods are used. Let us now learn the following methods:

1. Prime factorization method
2. Long division method
3. Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 25000 is broken down into its prime factors:

Step 1: Finding the prime factors of 25000 Breaking it down, we get 2 x 2 x 2 x 2 x 5 x 5 x 5 x 5 x 5: 2⁴ x 5⁵

Step 2: Now we found the prime factors of 25000. The second step is to make pairs of those prime factors. Since 25000 is a perfect square, we can pair the digits.

Therefore, calculating √25000 using prime factorization is possible: √25000 = √(2⁴ x 5⁵) = 2² x 5^(5/2) = 100 x 5^(1/2) = 100 x 5 = 500

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 25000, we group it as 250 and 00.

Step 2: Now we need to find n whose square is closest to the first group (250). We can say n as ‘15’ because 15 x 15 = 225, which is less than 250. Now the quotient is 15, and after subtracting, the remainder is 25.

Step 3: Now let us bring down 00, which is the new dividend. Add the old divisor with the same number 15 + 15, we get 30, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 30n as the new divisor; we need to find the value of n.

Step 5: The next step is finding 30n × n ≤ 2500. Let us consider n as 8; now 30 x 8 x 8 = 2400.

Step 6: Subtract 2500 from 2400; the difference is 100, and the quotient is 158.

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 10000.

Step 8: Now we need to find the new divisor, which is 316, because 316 x 3 = 948.

Step 9: Subtracting 948 from 1000, we get the result 52.

Step 10: Now the quotient is 158.11.

Step 11: Continue doing these steps until we get two numbers after the decimal point. So the square root of √25000 is 158.11.

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 25000 using the approximation method.

Step 1: We have to find the closest perfect square of √25000. The smallest perfect square less than 25000 is 22500, and the largest perfect square greater than 25000 is 25600. √25000 falls somewhere between 150 and 160.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square)

Using the formula (25000 - 22500) ÷ (25600 - 22500) = 0.8 Using the formula, we identified the decimal point of our square root.

The next step is adding the value we got initially to the decimal number, which is 150 + 0.8 = 150.8, so the approximate square root of 25000 is 150.8.

• Square root: A square root is the inverse of a square. Example: 4² = 16, and the inverse of the square is the square root, that is √16 = 4.

• 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 product of an integer multiplied by itself. Example: 16 is a perfect square because it is 4 x 4.

• Exponent: An exponent refers to the number of times a number is multiplied by itself. Example: 2^3 means 2 x 2 x 2.

• Approximation: Approximation is a method of finding a value that is close enough to the right answer, usually with some thought or consideration.