Square Root of 2268

The square root is the inverse of the square of the number. 2268 is not a perfect square. The square root of 2268 is expressed in both radical and exponential form. In the radical form, it is expressed as √2268, whereas (2268)^(1/2) in the exponential form. √2268 ≈ 47.62, 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 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 2268 is broken down into its prime factors:

Step 1: Finding the prime factors of 2268 Breaking it down, we get 2 × 2 × 3 × 3 × 3 × 3 × 7: 2^2 × 3^4 × 7

Step 2: Now we found out the prime factors of 2268. The second step is to make pairs of those prime factors. Since 2268 is not a perfect square, therefore the digits of the number can’t be grouped in pairs to find an exact square root.

However, we can estimate √2268 using the factors.

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 2268, we need to group it as 68 and 22.

Step 2: Now we need to find a number whose square is less than or equal to 22. We can say this number is 4 because 4 × 4 = 16, which is less than 22. Now the quotient is 4, and after subtracting 16 from 22, the remainder is 6.

Step 3: Bring down 68, making the new dividend 668. Add the old divisor (4) with the same number (4), getting 8, which will be our new divisor.

Step 4: The new divisor will be 8n, where we need to find n such that 8n × n ≤ 668.

Step 5: By trial, we find that 8 × 8 = 64, and 648 is close to 668. Subtracting gives us a remainder of 20, and the quotient becomes 46.

Step 6: Since the dividend is less than the divisor, we add a decimal point, allowing us to bring down two zeros, making the new dividend 2000.

Step 7: We find the new divisor to be 929, since 929 × 2 = 1858, which is less than 2000.

Step 8: Subtracting 1858 from 2000, we get a remainder of 142.

Step 9: Continue these steps until we get two numbers after the decimal point.

The square root of √2268 ≈ 47.62.

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 2268 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √2268. The smallest perfect square less than 2268 is 2025 (45²), and the largest perfect square greater than 2268 is 2304 (48²). √2268 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: (2268 - 2025) ÷ (2304 - 2025) ≈ 0.62. Using the formula, we identified the decimal point of our square root.

The next step is adding the initial integer value to the decimal number, which is 45 + 0.62 = 45.62.

Therefore, the square root of 2268 is approximately 47.62.

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

• Irrational number: An irrational number 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, but the positive square root is more commonly used in real-world applications. It is known as the principal square root.

• Prime factorization: Prime factorization is the process of expressing a number as the product of prime numbers.

• Decimal: A decimal is a number that contains both a whole number and a fractional part, represented with a decimal point, such as 7.86, 8.65, and 9.42.