Square Root of 2240

The square root is the inverse of the square of a number. 2240 is not a perfect square. The square root of 2240 can be expressed in both radical and exponential form. In radical form, it is expressed as √2240, whereas in exponential form it is expressed as (2240)^(1/2). √2240 ≈ 47.331, 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, for non-perfect square numbers, methods such as long division and approximation 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 2240 is broken down into its prime factors.

Step 1: Finding the prime factors of 2240 Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 5 x 7 = 2^5 x 5 x 7

Step 2: Now we have found the prime factors of 2240. The next step is to make pairs of those prime factors. Since 2240 is not a perfect square, the digits of the number cannot be grouped into pairs evenly.

Therefore, calculating 2240 using prime factorization is not straightforward.

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 2240, we need to group it as 40 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, and after subtracting 16 from 22, the remainder is 6.

Step 3: Now let us bring down 40, making the new dividend 640. Add the old divisor with the same number, 4 + 4, to get 8, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we have 8n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 8n × n ≤ 640. Let us consider n as 7, now 87 x 7 = 609.

Step 6: Subtract 609 from 640, the difference is 31, and the quotient is 47.

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 3100.

Step 8: Now find the new divisor, 94, because 944 x 4 = 3776, which is too large, so try 93, and 932 x 2 = 1864, which fits.

Step 9: Subtracting 1864 from 3100 results in 1236.

Step 10: Continue these steps until you get two numbers after the decimal point. Suppose there is no decimal value; continue until the remainder is zero.

So the square root of √2240 ≈ 47.33.

The approximation method is another method for finding square roots, and 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 2240 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √2240. The smallest perfect square below 2240 is 2025 (45^2) and the largest perfect square above 2240 is 2304 (48^2). √2240 falls somewhere between 45 and 48.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (greater perfect square - smallest perfect square).

Using the formula: (2240 - 2025) ÷ (2304 - 2025) = 215 ÷ 279 ≈ 0.77.

Add this value to the lower bound of our range: 45 + 0.77 = 45.77, which is an approximation.

So we refine further to 47.33.

• 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, which 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.

• Perfect square: A perfect square is an integer that is the square of an integer. For example, 9 is a perfect square because it is 3^2.

• Prime factorization: The process of expressing a number as the product of its prime factors.

• Long division method: A method used to find the square root of numbers, especially non-perfect squares, through a systematic division process.