Square Root of 22500

The square root is the inverse of the square of the number. 22500 is a perfect square. The square root of 22500 is expressed in both radical and exponential form. In the radical form, it is expressed as √22500, whereas (22500)^(1/2) in the exponential form. √22500 = 150, 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. Let's learn the methods to find the square root:

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 22500 is broken down into its prime factors:

Step 1: Finding the prime factors of 22500 Breaking it down, we get 2 x 2 x 3 x 3 x 5 x 5 x 5 x 5: 2² x 3² x 5⁴

Step 2: Now we found out the prime factors of 22500. The second step is to make pairs of those prime factors. Since 22500 is a perfect square, the digits of the number can be grouped in pairs.

Therefore, calculating √22500 using prime factorization gives us 150.

The long division method is particularly used for both perfect and non-perfect square numbers. 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 22500, we need to group it as 00, 50, and 22.

Step 2: Now we need to find n whose square is less than or equal to 22. We can say n as '4' because 4 x 4 = 16, which is less than 22. Now the quotient is 4 after subtracting 22 - 16, the remainder is 6.

Step 3: Now let us bring down 50, which is the new dividend. Add the old divisor (4) with itself to get 8, which will be our new divisor.

Step 4: The new divisor will be 80. Now we find n such that 8n x n ≤ 650. Let us consider n as 8, now 88 x 8 = 704.

Step 5: Subtracting 650 from 704 gives a remainder of 54.

Step 6: Bring down the next pair, 00, to make it 5400. The new divisor becomes 800. Find n such that 800n x n ≤ 5400. Here, n is 6, so 806 x 6 = 4836.

Step 7: Subtracting 4836 from 5400 gives a remainder of 564.

Step 8: Continue this process until no remainder is left (or until you reach a satisfactory decimal precision). The result is 150.

The approximation method is another method for finding the 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 22500 using the approximation method.

Step 1: Now we have to find the closest perfect square of √22500. The smallest perfect square less than 22500 is 19600 (140²), and the largest perfect square greater than 22500 is 25600 (160²). √22500 falls exactly at 150.

Step 2: Since 22500 is a perfect square, there is no need for further approximation. The square root of 22500 is exactly 150.

• Square root: A square root is a value that, when multiplied by itself, gives the original number. Example: 12² = 144, and the square root of 144 is √144 = 12.

• Rational number: A rational number is a number that can be expressed as a quotient or fraction p/q, where p and q are integers and q ≠ 0.

• Perfect square: A perfect square is a number that is the square of an integer. Example: 22500 is a perfect square because it is 150².

• Prime factorization: Prime factorization is the process of expressing a number as the product of its prime factors. Example: 22500 = 2² x 3² x 5⁴.

• Long division method: The long division method is a technique used to find the square root of a number by dividing it into groups and finding the quotient step by step.