Square Root of 1649

The square root is the inverse of the square of the number. 1649 is not a perfect square. The square root of 1649 is expressed in both radical and exponential form. In the radical form, it is expressed as √1649, whereas (1649)^(1/2) in the exponential form. √1649 ≈ 40.601 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 1649 is broken down into its prime factors.

Step 1: Finding the prime factors of 1649 1649 is not easily factorized into small primes, but through further testing, we find it as 1649 = 7 x 236 + 1, which does not break down into small prime factors easily.

Step 2: Since 1649 is not a perfect square, calculating its square root using prime factorization directly 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, we need to group the numbers from right to left. In the case of 1649, we need to group it as 49 and 16.

Step 2: Now we need to find n whose square is closest to 16. We can say n as '4' because 4 x 4 = 16. Now the quotient is 4, after subtracting 16 - 16, the remainder is 0.

Step 3: Now bring down 49, which is the new dividend. Add the old divisor with the same number 4 + 4 = 8, which will be our new divisor.

Step 4: The new divisor will be 8n as we continue. We need to find n such that 8n x n ≤ 49.

Step 5: By testing, we find n = 5 fits as 85 x 5 = 425.

Step 6: Subtract 49 from 425 and bring down the next pair of zeroes to continue the division.

Step 7: Continue the process by adding a decimal point and pairs of zeroes until the desired precision is achieved.

The approximate square root of 1649 is 40.601.

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 1649 using the approximation method.

Step 1: Find the closest perfect squares around 1649.

The smallest perfect square less than 1649 is 1600 (40^2) and the largest perfect square more than 1649 is 1681 (41^2).

√1649 falls between 40 and 41.

Step 2: Now apply the formula that is

(Given number - smallest perfect square) / (Greater perfect square - smallest perfect square)

Using the formula:

(1649 - 1600) / (1681 - 1600) = 49 / 81 ≈ 0.6049.

Adding this to the lower bound: 40 + 0.6049 ≈ 40.6049.

So the square root of 1649 is approximately 40.6049.

• Square root: A square root is the inverse of a square. For example, 5^2 = 25, and the inverse of the square is the square root, that is √25 = 5.
   
• 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 the positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as a principal square root.
   
• Long division method: A method used to find the square root of a non-perfect square by dividing the number into pairs and finding the square root step by step.
   
• Approximation method: A method to estimate the square root of a number by finding nearby perfect squares and interpolating between them.