Square Root of 1048

The square root is the inverse of the square of the number. 1048 is not a perfect square. The square root of 1048 is expressed in both radical and exponential form. In the radical form, it is expressed as √1048, whereas (1048)^(1/2) in the exponential form. √1048 ≈ 32.372, 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 1048 is broken down into its prime factors.

Step 1: Finding the prime factors of 1048 Breaking it down, we get 2 x 2 x 2 x 131: 2^3 x 131

Step 2: Now we have found the prime factors of 1048. The second step is to make pairs of those prime factors. Since 1048 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating √1048 using prime factorization is impossible.

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 1048, we need to group it as 10 and 48.

Step 2: Now we need to find n whose square is closest to 10. We can say n is ‘3’ because 3 x 3 = 9, which is less than or equal to 10. Now the quotient is 3, and after subtracting 9 from 10, the remainder is 1.

Step 3: Now let us bring down 48, which is the new dividend. Add the old divisor with the same number 3 + 3 to get 6, which will be our new divisor.

Step 4: Now we get 6n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 6n x n ≤ 148. Let us consider n as 2, now 6 x 2 x 2 = 144.

Step 6: Subtract 144 from 148, the difference is 4, and the quotient is 32.

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 zeros to the dividend. Now the new dividend is 400.

Step 8: Now we need to find the new divisor that is 644 because 644 x 6 = 3864.

Step 9: Subtracting 3864 from 4000, we get the result 136.

Step 10: Now the quotient is 32.3.

Step 11: Continue these steps until we get two numbers after the decimal point, or until the remainder is zero.

So the square root of √1048 is approximately 32.37.

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 1048 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √1048. The smallest perfect square less than 1048 is 1024, and the largest perfect square greater than 1048 is 1089. √1048 falls between √1024 (32) and √1089 (33).

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Using the formula: (1048 - 1024) ÷ (1089 - 1024) = 24 ÷ 65 ≈ 0.369 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 32 + 0.37 ≈ 32.37.

So the square root of 1048 is approximately 32.37.

• 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, the positive square root is more commonly used in the real world. This is known as the principal square root.
   
• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.
   
• Long division method: A traditional method used to find the square root of a number by dividing it step-by-step, especially useful for non-perfect squares.