Square Root of 1549

The square root is the inverse of the square of the number. 1549 is not a perfect square. The square root of 1549 is expressed in both radical and exponential form. In the radical form, it is expressed as √1549, whereas (1549)^(1/2) in the exponential form. √1549 ≈ 39.342, 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 1549 is broken down into its prime factors:

Step 1: Finding the prime factors of 1549 1549 is a prime number, so it cannot be broken down further.

Step 2: Since 1549 is not a perfect square, calculating its square root using prime factorization is not feasible.

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 1549, we need to group it as 49 and 15.

Step 2: Now we need to find n whose square is less than or equal to 15. We can say n as ‘3’ because 3 x 3 = 9, which is less than 15. Now the quotient is 3, and after subtracting 9 from 15, the remainder is 6.

Step 3: Now let us bring down 49, making it the new dividend. Add the old divisor with the same number 3 + 3 = 6, which will be our new divisor.

Step 4: The next step is finding a digit x such that 6x x x is less than or equal to 649. We find that x = 9 works because 69 x 9 = 621.

Step 5: Subtract 621 from 649, and the difference is 28. The quotient is now 39.

Step 6: Since the dividend is less than the divisor, we add a decimal point and two zeroes, making the new dividend 2800.

Step 7: Repeat the process to get more decimal places until the desired accuracy is achieved.

So the square root of √1549 ≈ 39.342

The approximation method is an easy method to find the square root of a given number. Now let us learn how to find the square root of 1549 using the approximation method.

Step 1: Find the closest perfect squares of √1549. The closest perfect squares are 1521 (39^2) and 1600 (40^2).

Step 2: 1549 is closer to 1521, so the square root is closer to 39.

Apply linear interpolation for better approximation.

(1549 - 1521) / (1600 - 1521) = (39.5 - 39) / (40 - 39)

Using this interpolation, we find √1549 ≈ 39.342, so the square root of 1549 is about 39.342.

• 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.
   
• Prime number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
   
• 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.
   
• Perfect square: A perfect square is an integer that is the square of an integer. For example, 1, 4, 9, 16, 25 are perfect squares.