Square Root of 1636

The square root is the inverse of the square of the number. 1636 is not a perfect square. The square root of 1636 is expressed in both radical and exponential form. In the radical form, it is expressed as √1636, whereas (1636)^(1/2) in the exponential form. √1636 ≈ 40.4475, 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 1636 is broken down into its prime factors:

Step 1: Finding the prime factors of 1636 Breaking it down, we get 2 x 2 x 409: 2^2 x 409

Step 2: Now we found out the prime factors of 1636. The second step is to make pairs of those prime factors. Since 1636 is not a perfect square, therefore, the digits of the number can’t be grouped in pairs.

Therefore, calculating 1636 using prime factorization is not feasible for finding an exact square root.

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 1636, we can group it as 16 and 36.

Step 2: Now we need to find n whose square is close to or less than 16. We can say n is '4' because 4 x 4 = 16. Now the quotient is 4 after subtracting 16 from 16, the remainder is 0.

Step 3: Now let us bring down 36, which is the new dividend. Add the old divisor with the same number, 4 + 4, to 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 ≤ 36; let us consider n as 4, now 8 x 4 = 32.

Step 6: Subtract 36 from 32, the difference is 4, and the quotient is 40.

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 400.

Step 8: Now we need to find the new divisor that is 80 because 804 ✖ 4 = 3216.

Step 9: Subtracting 3216 from 4000, we get the result 784.

Step 10: Now the quotient is 40.4.

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 √1636 is approximately 40.45.

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 1636 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1636. The smallest perfect square less than 1636 is 1600, and the largest perfect square greater than 1636 is 1681. √1636 falls somewhere between 40 and 41.

Step 2: Now we need to apply the formula:

(Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square).

Using the formula, (1636 - 1600) ÷ (1681 - 1600) = 36 ÷ 81 ≈ 0.444.

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 40 + 0.444 = 40.444, so the square root of 1636 is approximately 40.444.

• Square root: A square root is the inverse of a square. Example: 4² = 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 prominent due to its uses in the real world. This is why it is known as the principal square root.
   
• Divisor: A divisor is a number that divides another number wholly without leaving any remainder.
   
• Perfect square: A number that is the square of an integer. For example, 16 is a perfect square because it is 4².