Square Root of 1348

The square root is the inverse operation of squaring a number. 1348 is not a perfect square. The square root of 1348 is expressed in both radical and exponential form. In the radical form, it is expressed as √1348, whereas (1348)^(1/2) in the exponential form. √1348 ≈ 36.719, 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 prime factorization of a number is its expression as a product of prime factors. Now let us look at how 1348 is broken down into its prime factors:

Step 1: Finding the prime factors of 1348

Breaking it down, we get 2 x 2 x 337: 2^2 x 337^1

Step 2: Now that we have found the prime factors of 1348, we attempt to make pairs of those prime factors. Since 1348 is not a perfect square, calculating √1348 using prime factorization is not straightforward.

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: Start by grouping the digits of 1348 from right to left in pairs. We have 13 and 48.

Step 2: Find the largest number whose square is less than or equal to 13. This number is 3, since 3 x 3 = 9.

Step 3: Subtract 9 from 13 to get 4. Bring down the next pair, 48, to have 448.

Step 4: Double the divisor (3) to get 6, and set up a new divisor as 6n, where n is a digit that, when placed next to 6, forms a number that fits into 448.

Step 5: Determine n such that 6n x n is less than or equal to 448. Let n be 7, as 67 x 7 = 469 is too large, and 66 x 6 = 396 fits.

Step 6: Subtract 396 from 448 to get 52.

Step 7: Bring down two zeros to make it 5200, and repeat the process to get more decimal places.

Step 8: Continue the process to get a more precise value, yielding approximately √1348 ≈ 36.719.

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 1348 using the approximation method:

Step 1: Identify the closest perfect squares around 1348. The smallest perfect square less than 1348 is 1296 (36^2) and the largest perfect square greater than 1348 is 1369 (37^2).

Step 2: Since √1348 is between 36 and 37, we can use the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square)

Step 3: Apply the formula: (1348 - 1296) / (1369 - 1296) = 52 / 73 ≈ 0.712 Step 4: Add this decimal to the smaller square root: 36 + 0.712 = 36.712. So √1348 ≈ 36.719.

• Square root: A square root of a number is a value that, when multiplied by itself, gives the original number. Example: 4^2 = 16, and the square root of 16 is √16 = 4.

• Irrational number: An irrational number is a number that cannot be written as a simple fraction - it has a non-repeating, non-terminating decimal expansion.

• Perfect square: A perfect square is an integer that is the square of another integer. For example, 36 is a perfect square because it is 6^2.

• Long division method: A technique used to find the square root of a number by dividing and averaging, typically used for non-perfect squares.

• Radical notation: The expression of a square root using the radical sign (√). For example, √25 = 5.