Square Root of 2262

The square root is the inverse of the square of the number. 2262 is not a perfect square. The square root of 2262 is expressed in both radical and exponential form. In the radical form, it is expressed as √2262, whereas (2262)^(1/2) in the exponential form. √2262 ≈ 47.5533, 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, the prime factorization method is not used; instead, 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 2262 is broken down into its prime factors.

Step 1: Finding the prime factors of 2262 Breaking it down, we get 2 x 3 x 3 x 3 x 3 x 7: 2^1 x 3^4 x 7^1

Step 2: Now that we have found the prime factors of 2262, the second step is to make pairs of those prime factors. Since 2262 is not a perfect square, therefore the digits of the number can’t be grouped into pairs.

Therefore, calculating 2262 using prime factorization is impractical for finding the exact square root.

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 2262, we need to group it as 22 and 62.

Step 2: Now we need to find n whose square is closest to 22. We can say n is ‘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 62, which is the new dividend. Add the old divisor with the same number: 4 + 4, we get 8, which will be our new divisor.

Step 4: The new divisor will be the sum of the current divisor and the quotient, so 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 ≤ 662. Let us consider n as 8, now 88 x 8 = 704 which is greater than 662, so we take n as 7.

Step 6: Subtract 662 from 609 (87 x 7 = 609), the difference is 53, 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 5300.

Step 8: Now we need to find the new divisor that is 951 because 951 x 5 = 4755.

Step 9: Subtracting 4755 from 5300, we get the result 545.

Step 10: Now the quotient is 47.5.

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

So the square root of √2262 is approximately 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 2262 using the approximation method.

Step 1: Now we have to find the closest perfect square of √2262. The smallest perfect square of 2262 is 2025, and the largest perfect square is 2304. √2262 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).

Going by the formula (2262 - 2025) ÷ (2304 - 2025) = 237 ÷ 279 ≈ 0.85.

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 47 + 0.55 = 47.55.

So the square root of 2262 is approximately 47.55.

• 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 a positive square root that has more prominence due to its uses in the real world. That is why it is also known as the principal square root.

• Long division method: A method used to find the square root of non-perfect squares by pairing digits from right to left and estimating successive digits of the quotient.

• Approximation method: A method for estimating the square root by finding the nearest perfect squares and interpolating to find an approximate value.