Square Root of 2260

The square root is the inverse of the square of a number. 2260 is not a perfect square. The square root of 2260 is expressed in both radical and exponential form. In the radical form, it is expressed as √2260, whereas (2260)^(1/2) in the exponential form. √2260 ≈ 47.549, 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 the long-division method and approximation method 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 2260 is broken down into its prime factors.

Step 1: Finding the prime factors of 2260 Breaking it down, we get 2 x 2 x 5 x 113: 2^2 x 5^1 x 113^1

Step 2: Now we found the prime factors of 2260. The second step is to make pairs of those prime factors. Since 2260 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating √2260 using prime factorization directly is not feasible.

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 2260, we need to group it as 60 and 22.

Step 2: Now, we need to find n whose square is 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 60, making the new dividend 660. 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 get 8n as the new divisor; we need to find the value of n.

Step 5: The next step is finding 8n x n ≤ 660. Let's consider n as 7, so 87 x 7 = 609.

Step 6: Subtract 609 from 660; the difference is 51, 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 allows us to add two zeros to the dividend. Now the new dividend is 5100.

Step 8: Now we need to find the new divisor, 945, because 945 x 5 = 4725.

Step 9: Subtracting 4725 from 5100 gives us 375.

Step 10: Now the quotient is 47.5.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal value, continue till the remainder is zero.

So the square root of √2260 ≈ 47.55.

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 2260 using the approximation method.

Step 1: Now we have to find the closest perfect squares around √2260. The smallest perfect square smaller than 2260 is 2025, and the largest perfect square larger than 2260 is 2304. √2260 falls somewhere between 45 and 48.

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 (2260 - 2025) ÷ (2304 - 2025) = 0.836.

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 45 + 0.836 = 45.836.

So the square root of 2260 is approximately 45.836.

• 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 always the positive square root that is often used in practical applications, known as the principal square root.

• Prime factorization: The process of breaking down a composite number into its prime factors. For example, the prime factorization of 18 is 2 x 3^2.

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