Square Root of 2048

The square root is the inverse of the square of the number. 2048 is not a perfect square. The square root of 2048 is expressed in both radical and exponential form. In the radical form, it is expressed as √2048, whereas (2048)^(1/2) in the exponential form. √2048 ≈ 45.2548, 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 2048 is broken down into its prime factors.

Step 1: Finding the prime factors of 2048 Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2: 2^11

Step 2: Now we found out the prime factors of 2048. The second step is to make pairs of those prime factors. Since 2048 is not a perfect square, therefore the digits of the number can’t be grouped in pairs. Therefore, calculating 2048 using prime factorization is impossible for a perfect square.

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 2048, we need to group it as 48 and 20.

Step 2: Now we need to find n whose square is ≤ 20. We can say n as '4' because 4 x 4 = 16 is lesser than or equal to 20. Now the quotient is 4 after subtracting 16 from 20, the remainder is 4.

Step 3: Now let us bring down 48 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 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 × n ≤ 448. Let us consider n as 5, now 85 x 5 = 425.

Step 6: Subtract 448 from 425, the difference is 23, and the quotient is 45.

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 2300.

Step 8: Now we need to find the new divisor that is 452 because 452 x 5 = 2260.

Step 9: Subtracting 2260 from 2300, we get the result 40.

Step 10: Now the quotient is 45.2.

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

So the square root of √2048 is approximately 45.25.

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 2048 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √2048. The smallest perfect square less than 2048 is 2025 and the largest perfect square greater than 2048 is 2116. √2048 falls somewhere between 45 and 46.

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 (2048 - 2025) ÷ (2116 - 2025) ≈ 0.253.

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 + 0.253 = 45.253, so the square root of 2048 is approximately 45.253.

• 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, q is not equal to zero and p and q are integers.
   
• Radical form: The radical form is a way of expressing the square root using the radical sign (√), for example, √2048.
   
• Decimal approximation: A decimal approximation is an estimate of an irrational number to a certain number of decimal places, for example, √2048 ≈ 45.2548.
   
• Prime factorization: Prime factorization is expressing a number as a product of its prime factors, for example, 2048 = 2^11.