Square Root of 1564

The square root is the inverse of the square of the number. 1564 is not a perfect square. The square root of 1564 is expressed in both radical and exponential form. In the radical form, it is expressed as √1564, whereas (1564)^(1/2) in the exponential form. √1564 ≈ 39.536, 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 1564 is broken down into its prime factors.

Step 1: Finding the prime factors of 1564 Breaking it down, we get 2 x 2 x 19 x 41: 2^2 x 19 x 41

Step 2: Now we found out the prime factors of 1564. The second step is to make pairs of those prime factors. Since 1564 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating 1564 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: To begin with, we need to group the numbers from right to left. In the case of 1564, we need to group it as 64 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 3^2 = 9, which is lesser than or equal to 15. Now the quotient is 3, after subtracting 9 from 15 the remainder is 6.

Step 3: Now let us bring down 64 which is the new dividend. Add the old divisor with the same number 3 + 3 we get 6 which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 6n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 6n × n ≤ 664. Let us consider n as 9, now 69 × 9 = 621.

Step 6: Subtract 664 from 621, the difference is 43, and the quotient is 39.

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 4300.

Step 8: Now we need to find the new divisor. Let us consider 395 because 395 × 5 = 1975.

Step 9: Subtracting 1975 from 4300 we get the result 2325.

Step 10: Now the quotient is 39.5

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 √1564 ≈ 39.54

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 1564 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1564.

The smallest perfect square less than 1564 is 1521 (39^2), and the largest perfect square greater than 1564 is 1600 (40^2).

√1564 falls somewhere between 39 and 40.

Step 2: Now we need to apply the formula that is

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

Going by the formula (1564 - 1521) ÷ (1600 - 1521) = 43 ÷ 79 ≈ 0.544

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 39 + 0.544 = 39.544, so the square root of 1564 is approximately 39.544

• 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, 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.
   
• Factorization: Factorization is the process of breaking down a number into its prime factors.
   
• Approximation method: This is a method to estimate the value of a number, especially useful when calculating the square roots of non-perfect squares.